The total number of locks is given as follows:
941,094 locks.
What is the permutation formula?The number of possible permutations of x elements from a set of n elements is given by:
\(P_{(n,x)} = \frac{n!}{(n-x)!}\)
The permutation formula is used for this problem as the order in which the numbers are chosen is important.
3 numbers are chosen from a set of 99, hence the total number of locks is given as follows:
P(99,3) = 99!/96! = 99 x 98 x 97 = 941,094 locks.
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The value of the _____ is used to estimate the value of the population parameter.
The value of the sample statistic is used to estimate the value of the population parameter.
What estimates the value of the population parameter?The population parameter is found by using the population statistic to predict the population estimate.
This is because the sample statistic is based on the sample produced from the population so it can therefore be used to find out more on the population.
Options for this question are:
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Given r(x) =x' - 4x2 + 4x - 6, find the value of r(2). What does your answer tell you about x - 2 as a factor of r(x)? Explain.
We can conclude that the value of r(2) which is -6 and x - 2 is not the factor for r(x).
What is Factor?A function that accepts a real number as input, typically represented by the variable x and outputs another real number, typically the value of the function denoted by f is known as a real-valued function of a real variable (x).Any polynomial that is divisible by the polynomial P(x) is a factor of that polynomial (x).So, we know that the equation is:
r(x)=x^3-4x^2+4x-6:Now, calculate the equation when x = 2.
r(2)=2^3-4(2)^2+4(2)-6r(2)=8-16+8-6r(2)=-6Now, we can conclude that:
As we see that x=2 is not equal to zero.Implies x-2 does not divide r(x) evenly.Thus x-2 is not a factor of given r(x).Therefore, we can conclude that the value of r(2) which is -6 and x - 2 is not the factor for r(x).
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PLEASE HELP AND EXPLAIN!
A. y=60x
B. y=80x
C. y=10x
D. y= 6x
Answer:
C).
Step-by-step explanation:
One of the points on the graph that the path of bike b crosses is (6,60).
When you divide this to find the unit rate you get 10/1 therefore the equation of bike b= y+10X
The weights of a certain brand of candies are normally distributed with a mean weight of 0.8545 g and a standard deviation of 0.0517 g. A sample of these candies came from a package containing 458 candies, and the package label stated that the net weight is 391.0 g. (If every package has 458 candies, the mean weight of the candies must exceed -=0.8537 g for the net contents to weigh at least 391.0 g.) 391.0 458 Tre a. If 1 candy is randomly selected, find the probability that it weighs more than 0.8537 g The probability is 0.5062 (Round to four decimal places as needed.) b. If 458 candies are randomly selected, find the probability that their mean weight is at least 0.8537 g The probability that a sample of 458 candies will have a mean of 0.8537 g or greater is 0 (Round to four decimal places as needed.)
a. the probability that a randomly selected candy weighs more than 0.8537 g is approximately 0.5062.
b. the probability that a sample of 458 candies will have a mean weight of at least 0.8537 g is approximately 0.4920.
a. To find the probability that a randomly selected candy weighs more than 0.8537 g, we can use the z-score and the standard normal distribution.
Given:
Mean weight (μ) = 0.8545 g
Standard deviation (σ) = 0.0517 g
We need to find the probability P(X > 0.8537), where X is the weight of a randomly selected candy.
First, let's calculate the z-score for 0.8537 g:
z = (x - μ) / σ
z = (0.8537 - 0.8545) / 0.0517
z ≈ -0.0155
Using the standard normal distribution table or a calculator, we can find the probability corresponding to a z-score of -0.0155, which is approximately 0.5062.
Therefore, the probability that a randomly selected candy weighs more than 0.8537 g is approximately 0.5062.
b. To find the probability that a sample of 458 candies will have a mean weight of at least 0.8537 g, we need to calculate the sampling distribution of the sample mean.
Given:
Sample size (n) = 458
Mean weight (μ) = 0.8545 g
Standard deviation (σ) = 0.0517 g
The sample mean follows a normal distribution with the same mean as the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size (σ/√n).
Standard deviation of the sample mean (σ/√n) = 0.0517 / √458 ≈ 0.002415
To find the probability P(\(\bar{X}\) ≥ 0.8537), where \(\bar{X}\) is the mean weight of the sample of 458 candies:
Using the z-score formula:
z = (\(\bar{X}\) - μ) / (σ/√n)
z = (0.8537 - 0.8545) / 0.002415
z ≈ -0.0331
Using the standard normal distribution table or a calculator, the probability corresponding to a z-score of -0.0331 is approximately 0.4920.
Therefore, the probability that a sample of 458 candies will have a mean weight of at least 0.8537 g is approximately 0.4920.
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Two numbers are in the ration of 3:4. If their sum is 133. Find the numbers.
Answer:
57 and 76
Step-by-step explanation:
the ratio = 3 : 4
=> the sum : 3+4=7 ( equivalent with 133)
so the numbers are :
3/7 × 133 = 57 and
4/7 × 133 = 76.
Answer:
The numbers are x = 57 and y = 76.
Step-by-step explanation:
Represent the numbers by x and y.
x 3
Then --- = -----
y 4
Also, x + y = 133. We can solve this for either x or y and substitute the result into the above equation. Solving for y, we get 133 - x = y, and so the above equation becomes
x 3
--------- = ------
133 - x 4
Cross multiplying yields
4x = 399 - 3x, or 7x = 399, or x = 57. Then, since y = 133 - x, y = 76.
The numbers are x = 57 and y = 76.
Ayudaaaaaaaaaaaaaaaaaaaaaaaaaaa plz xd
Answer:
e = 3
Step-by-step explanation:
is what my brother said
if its right can i maybe have brainliest because i am trying so so hard to level up
FAST PLEASE
What is the simplified expression for 4 to the power of negative 3 multiplied by 3 to the power of 4 multiplied by 4 to the power of 2 whole over 3 to the power of 5 multiplied by 4 to the power of negative 2 ?
3 over 4
4 over 3
4 to the power of 2 over 3
3 to the power of 3 over 4
Answer:
4/3
Step-by-step explanation:
You can simplify the exponent expression using exponent rules. The rules are:
A positive exponent is the number of times the base multiplies by itself.
A negative exponent is the number of times the base divides itself.
Multiplying same bases with exponents is simplified by adding the exponents.
Dividing same bases with exponents is simplified by subtracting the exponents.
A zero exponent always evaluates as 1.
The expression \frac{4^{-3}3^44^2}{3^54^{-2}}
3
5
4
−2
4
−3
3
4
4
2
can be simplified first using the negative exponent rule to move base with negative exponent to the other part of the fraction.
\frac{4^{-3}3^44^2}{3^54^{-2}} = \frac{4^{2}3^44^2}{3^54^{3}}
3
5
4
−2
4
−3
3
4
4
2
=
3
5
4
3
4
2
3
4
4
2
Now use the multiplication rule to simplify numerator and denominator.
\frac{4^{2}3^44^2}{3^54^{3}} = \frac{4^{2+2}3^4}{3^54^{3}} = \frac{4^{4}3^4}{3^54^{3}}
3
5
4
3
4
2
3
4
4
2
=
3
5
4
3
4
2+2
3
4
=
3
5
4
3
4
4
3
4
Finally, use the division rule to reduce the fraction.
\frac{4^{4}3^4}{3^54^{3}}= 4^{4-3} 3^{4-5} = 4*3^{-1} = \frac{4}{3}
3
5
4
3
4
4
3
4
=4
4−3
3
4−5
=4∗3
−1
=
3
4
PLS HELP THOU STRUGGLING STUDENT!!! X0
Answer:
C. Neither
Step-by-step explanation:
This is because the picture is showing both Exponential Growth AND Exponential Decay. As a result, it is both, but that is not an option, so neither is the answer. Hope it helps, and thank you for choosing Brainly for your question :D
Answer:
exponential decay
Step-by-step explanation:
We are multiplying by 1/3 each time
81 * 1/3 = 27
27*1/3 = 9
9*1/3 = 3
This is getting smaller so it is exponential decay
How do you solve for 12a. and what is 12a.?
Answer:
y=20*3^x
Step-by-step explanation:
y=20*3^x
Which statements are true about the components of a circle? Check all that apply.
Answer:
circles are measur by 360°.
circles are similar triangle.
circles are two radis.
circles are one diameter.
Answer:
1,3,4
Step-by-step explanation:
Bryan has a collection of nickels and quarters for a total value of $4.90. If he has a total of 42 coins, how many of each kind are there?
Answer:
Hope this helps!
Step-by-step explanation:
Bryan has a collection of nickels and quarters for a total value of 4.90.if she has 42 coins total how many of each kind are there
what is 0.000036÷180 in standard form
Answer: 36/1000000 is the pq form of the given no..
Step-by-step explanation:
The standard form of 0.000036÷180 is 2 × \(10^{-7}\) .
How do you express a number in standard form?Any number that we can write as a decimal number, between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form. 1.23 × \(10^{8}\); If you observe carefully, 1.23 is a decimal number between 1.0 and 10.0 and so we have the standard form of 123,000,000 as 1.23 × \(10^{8}\).
According to the question
0.000036÷180 in standard form
Now,
0.000036÷180
= \(\frac{0.000036}{180}\)
= 0.0000002
= \(\frac{2}{10000000}\)
= 2 × \(10^{-7}\) is the standard form .
Hence, the standard form of 0.000036÷180 is 2 × \(10^{-7}\) .
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The ratio of pretzels to crackers in a snack mix recipe is 11:9. What percent of the mix is
crackers?
Answer:
45%
Step-by-step explanation:
You would add 11 and 9 together to find the mix number, which would be 20. Then, you see that the crackers is 9. So, you would do 9 divided by 20 (to find the percentage). 9 divided by 20 is 0.45
0.45 as a percent is 45%.
A.y=-x+0.5
B.y=-x-0.5
C.y=x+0.5
D.y=x-0.5
Answer:
D
Step-by-step explanation:
f(x)=18x+11, find f(5)
Answer:
\(\boxed {\tt f(5)=101}\)
Step-by-step explanation:
We are given the function:
\(f(x)=18x+11\)
and asked to find f(5). Therefore, we must substitute 5 in for each \(x\).
\(f(5)=18(5)+11\)
Solve according to PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Multiply 18 and 5.
\(f(5)=90+11\)
Add 90 and 11.
\(f(5)=101\)
f(5) is equal to 101.
3/2 (x-5) - 3/2 = 9/2 x solve and verify
Answer:
x = -2
Step-by-step explanation:
3/2 (x-5) - 3/2 = 9x/2
3x/2 - 15/2 - 3/2 = 9x/2 ... alldenominator
are 2 so we can multiply all term by 2
2(3x/2 - 15/2 - 3/2 = 9x/2)
3x - 15 - 3 = 9x ... simplify it
3x - 12 = 9x
-12 = 9x - 3x ... take 3x to the left
-12 = 6x ... simplify
6x/6 = -12/6 ... dividing both side by 6
x = -2 ... simplify and solve x
rite the maclaurin series for f(x)=8x2sin(7x)f(x)=8x2sin(7x) as [infinity]
∑ cn x^n
n=0 find the following coefficients.
The Maclaurin series for f(x) is f(x) = 16x^2 - 914.6667x^3 + O(x^4).
To find the Maclaurin series for the function f(x) = 8x^2sin(7x), we need to compute its derivatives and evaluate them at x=0:
f(x) = 8x^2sin(7x)
f'(x) = 16xsin(7x) + 56x^2cos(7x)
f''(x) = 16(2cos(7x) - 49xsin(7x)) + 112xcos(7x)
f'''(x) = 16(-98sin(7x) - 343xcos(7x)) + 112(-sin(7x) + 7xcos(7x))
f''''(x) = 16(-2401cos(7x) + 2401xsin(7x)) + 784xsin(7x)
At x=0, all the terms with sin(7x) vanish, and we are left with:
f(0) = 0
f'(0) = 0
f''(0) = 32
f'''(0) = -5488
f''''(0) = 0
Thus, the Maclaurin series for f(x) is:
f(x) = 32x^2 - 2744x^3 + O(x^4)
We can also find the coefficients directly by using the formula:
cn = f^(n)(0) / n!
where f^(n)(0) is the nth derivative of f(x) evaluated at x=0. Using this formula, we get:
c0 = f(0) / 0! = 0
c1 = f'(0) / 1! = 0
c2 = f''(0) / 2! = 32 / 2 = 16
c3 = f'''(0) / 3! = -5488 / 6 = -914.6667
c4 = f''''(0) / 4! = 0 / 24 = 0
Therefore, the Maclaurin series for f(x) is:
f(x) = 16x^2 - 914.6667x^3 + O(x^4)
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help please i need the answer to this
Answer:
140 degrees
Step-by-step explanation:
first start by making the given angles equal each other because they are congruent.
2x+28=6x+4
Subtract 4 on both sides
2x+24=6x
subtract 2x on both sides
24=4x
divide both sides by 4 to get x
24/4=4x/4
6=x
next to find the actual number of the angle, plug 6 back into either one of the equations. Im gonna use the first one.
2x + 28
2(6) + 28
12+28= 40
Now that we have the number, we already know that a full circle is 360 degrees so half a circle is 180.
Take 40 and subtract from 180
180-40= 140
Well, m∠1 and m∠3 are congruent by the Vertical Angle Congruent Theorem (VACT). This means that because the angles are opposite of each other, their angles are the same because they are formed by the same two intersecting lines.
Because of this, we can set m∠1 and m∠3 equal to each other:
m∠1 = m∠3
(2x + 28) = (6x + 4)
2x + 28 = 6x + 4
-4x + 28 = 4
-4x = -24
-x = -6
x = 6
Knowing that x equals 6, we can use this to find m∠2. By the Supplementary Angles Theorem (SAT), angles that are on the same line add up to equal 180°. To use the SAT, we need to plug x in to find m∠1 or m∠3, because they are both the same:
x = 6
m∠1 = (2x + 28)
m∠1 = 2(6) + 28
m∠1 = 12 + 28
m∠1 = 40°
m∠3 also equals 40°.
Now that we have the value for ∠1 (or ∠3), we can use the SAT:
m∠2 + m∠3 = 180 ← You could also use m∠1, because they both equal the same!
m∠2 + 40 = 180
m∠2 = 140°
So, the measure of ∠2 is 140°.
If you have any questions, feel free to ask! :)
prove that √2 + √3 is irrational
Step-by-step explanation:
Let √2 + √3 be a / b.
a and b are co - primes.
a and b have 1 as a common factor.
√2 + √3 = a / b
( √2 + √3 )^2 = ( a / b )^2
(√2)^2 + (√3)^2 + 2.√2.√3 = a^2 / b^2
2 + 3 + 2.√2.√3 = a^2 / b^2
5 + 2.√2.√3 = a^2 / b^2
2.√2.√3 = ( a^2 / b^2 ) - 5
2.√2.√3 = ( a^2 - 5b^2 ) / b^2
√2.√3 = ( a^2 - 5b^2 ) / 2b^2
a and b are not co - primes, as they more than 1 as a common factor.
So,
√2 + √3 is irrational.
Hence, proved.
What is the solution to the equation x - 12 = 36?x = 24x = 48x = 36x = -3
ANSWER :
x = 48
EXPLANATION :
From the problem, we have :
\(x-12=36\)add 12 to both sides :
\(\begin{gathered} x-12+12=36+12 \\ x=48 \end{gathered}\)Hbbbbbbbbmmmmmhhehsjsdjs
Diinineinnneeeeeeeeeeeeeeeeeeeee
QuestionSolve the inequality 37v < -18 and write the solution in interval notation, using improper fractions if necessary.
A line that has a slope of 1/2 and passes through the point (-4,7)
Answer:
y= ½x+9
Step-by-step explanation:
Use desmos makes it easier
how much metal is needed to create a trash can with a height of 4 feet and base radius of 1 foot
Answer:
A ≈ 31.42
Step-by-step explanation:
Define convenience purchases, shopping purchases, and specialty purchases. Describe three specific brand name products in the consumer marketplace today that would correspond to these three types of purchases.
Convenience purchase: Coca-Cola. Shopping purchase: Apple iPhone. Specialty purchase: Rolex. These brand name products correspond to their respective purchase types based on convenience, shopping involvement, and specialty appeal in the consumer marketplace.
Convenience purchases refer to low-involvement purchases made by consumers for everyday items that are readily available and require minimal effort to obtain. These purchases are typically driven by convenience and habit rather than extensive consideration or brand loyalty.
Shopping purchases involve higher involvement and more deliberate decision-making. Consumers invest time and effort in comparing options, seeking the best value or quality, and may consider multiple brands before making a purchase. These purchases often involve durable goods or products that require more consideration.
Specialty purchases are distinct and unique purchases that cater to specific interests, preferences, or hobbies. These purchases are driven by passion, expertise, and a desire for premium or specialized products. Consumers are often willing to invest more in these purchases due to their unique features, quality, or exclusivity.
Three specific brand name products in the consumer marketplace that correspond to these types of purchases are
Convenience Purchase: Coca-Cola (Soft Drink)
Coca-Cola is a widely recognized brand in the beverage industry. It is readily available in various sizes and formats, making it a convenient choice for consumers seeking a refreshing drink on the go.
With its widespread availability and strong brand presence, consumers often make convenience purchases of Coca-Cola without much thought or consideration.
Shopping Purchase: Apple iPhone (Smartphone)
The Apple iPhone is a popular choice for consumers when it comes to shopping purchases. People invest time researching and comparing features, pricing, and user reviews before making a decision.
The shopping process involves considering various smartphone brands and models to ensure they select a product that meets their specific needs and preferences.
Specialty Purchase: Rolex (Luxury Watches)
Rolex is a well-known brand in the luxury watch industry and represents specialty purchases. These watches are associated with high-quality craftsmanship, precision, and exclusivity.
Consumers who are passionate about luxury watches and seek a premium product often consider Rolex due to its reputation, heritage, and unique features. The decision to purchase a Rolex involves a significant investment and is driven by the desire for a prestigious timepiece.
These examples illustrate how different types of purchases align with specific brand name products in the consumer marketplace, ranging from convenience-driven choices to more involved shopping decisions and specialty purchases driven by passion and exclusivity.
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7) The principal at a high school is planning a concert to raise money for the music programs. He determines
the profit p from ticket sales depends on the price of a ticket according to the equation
p = -200t2 + 3600t - 6400. All amounts are in dollars. If the goal is to raise $8500, what is the smallest
amount the school should charge for a ticket to the concert?
Answer:
The smallest amout that the school should charge is $4.70.
Step-by-step explanation:
p = -200t2 + 3600t - 6400
p = -200 x 4.7 x 2 + 3600 x 4.7 - 6400
p = -1880 + 16920 - 6400
p = 15120 - 6400
p = 8720
Answer: 6, 45 $
Step-by-step explanation:
\(\displaystyle -200t^2+3600t-6400=8500 \ \ |\div100\\\\ -2t^2+36t-64-85 =0 \ \ |\cdot (-1) \\\\ 2t^2-36t+149=0 \\\\ \rm D=36^2-4\cdot2\cdot 149=104 \\\\ x_1=\frac{36+2\sqrt{26} }{4} \approx11,54 \\\\\\ x_2=\frac{36-2\sqrt{26} }{4} \approx 6,45 \\\\ x_2<x_1 \\\\ Then :\\\\ Answer : 6,45 \$\)
1. Please use Temperature of 50 C for the constant temperature diagram and use 5 bar for the constant pressure diagram. If these variables don't work with your system, please inform us. 2. You need to - find the bubble and dew points at 50% feed composition - find the vapor pressure of the mixture pure components
The bubble point and dew points at 50% feed composition are 324.04 K and 189.54 K. The vapor pressure of the mixture pure components are 3.21 bar and 0.92 bar.
Calculating the bubble and dew points at 50% feed composition
The bubble point is the temperature at which the first vapor bubbles form in a liquid mixture. The dew point is the temperature at which the first liquid droplets form in a vapor mixture.
The bubble and dew points at 50% feed composition can be calculated using the following equations:
Bubble point = \(T_c\) * \((1 - X_f)^2\)
Dew point = \(T_c\) * \((X_f)^2\)
where:
\(T_c\) is the critical temperature of the mixture
\(X_f\) is the mole fraction of the feed component
The critical temperature of the mixture can be calculated using the following equation:
\(T_c\) = \((T_{c_1} * T_{c_2})^{1/2\)
where:
\(T_{c_1\) is the critical temperature of the first component
\(T_{c_2\) is the critical temperature of the second component
In this case, the critical temperature of the mixture is:
\(T_c\) = \((304.13 * 407.3)^{1/2\) = 379.08 K
The mole fraction of the feed component is 0.5. Therefore, the bubble point and dew points at 50% feed composition are:
Bubble point = 379.08 * \((1 - 0.5)^2\) = 324.04 K
Dew point = 379.08 * \((0.5)^2\) = 189.54 K
Calculating the vapor pressure of the mixture pure components
The vapor pressure of a pure component is the pressure at which the liquid and vapor phases of the component are in equilibrium.
The vapor pressure of the mixture pure components can be calculated using the following equation:
\(P_i\) = \(P_c\) * \((X_i / T_c)^{0.5\)
where:
\(P_i\) is the vapor pressure of the pure component i
\(P_c\) is the critical pressure of the pure component i
\(X_i\) is the mole fraction of the pure component i
\(T_c\) is the critical temperature of the mixture
In this case, the critical pressure of the first component is 220.6 bar and the critical pressure of the second component is 73.8 bar. Therefore, the vapor pressure of the mixture pure components are:
\(P_1\) = 220.6 * \((0.5 / 379.08)^{0.5\) = 3.21 bar
\(P_2\) = 73.8 * \((0.5 / 379.08)^{0.5\) = 0.92 bar
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Jimmy wants to draw a 50° angle. Fill in the missing step to make sure his drawing is completed in the correct order: (3 points)
Step 1: He draws a ray.
Step 2: He lines up his protractor with the baseline on the ray and the origin of the protractor on the vertex.
Step 3: ________
Step 4: He uses the bottom of the protractor to draw a straight line connecting the mark he made with the vertex.
Step 5: He draws arrows on both ends to make them rays.
a
He lines up his ruler with the ray and the zero on the ruler.
b
He finds 50° on the protractor and draws a mark on the paper.
c
He lines up his protractor with a straight line and draws a mark on the paper.
d
He draws parallel lines.
The step that is missing for Jimmy to draw a 50° angle is "He finds 50° on the protractor and draws a mark on the paper" (option B).
What is the next step Jimmy needs to complete?According to the process described, Jimmy has already draw the first ray that is going to work as the baseline for him to draw the angle he requires. After this, he will need to line up his protactor with this ray (step 1).
Then the next step would be to use a pencil or similar material to make a mark in the 50°, this is an important step as it guarantees the angle is correct before he draw the line.
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SHOW YOUR WORK PLEASE
You would like to purchase the car in 2 years. How much money will you need to invest at a 3. 3% interest rate compounded annually in order to have $9500 in 2 years? Use the compound interest formula A = P (1 + i)n. (Round final answer to the nearest cent, but otherwise don’t round any intermediate values)
$8,905.26 should be invested at a 3.3% interest rate compounded annually in order to have $9500 in 2 years is the correct answer.
Given, Initial Investment P = ? Interest Rate i = 3.3% = 0.033 (Annual rate) Time n = 2 years (Compounded annually) Total Amount after 2 years A = $9,500
We have to use the compound interest formula to find the initial investment.
Compound Interest formula: A = P (1 + i)n where A = Final amount P = Principal amount i = Annual interest rate (in decimal form) n = Number of years
Let's substitute the given values in the compound interest formula, we get; 9500 = P (1 + 0.033)2=> 9500 = P (1.033)2=> 9500 = 1.067P
Now, divide both sides of the equation by 1.067:=> P = 9500 / 1.067P = $8,905.26
Hence, $8,905.26 should be invested at a 3.3% interest rate compounded annually in order to have $9500 in 2 years.
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An algebra class has 12 students and 12 desks. For the sake of variety, students change the seating arrangement each day. How many days must pass before the class must repeat a seating arrangement?
__________ days must pass before a seating arrangement is repeated.
Suppose the desks are arranged in rows of 3. How many seating arrangements are there that put Larry, Moe, & Curly in the front seats?
There are ______ seating arrangements that put them in the front seats.
15.S={y∣y∈Nand103≤y<119}S={y∣y∈ℕand103≤y<119}
SS has ______ subsets.
SS has ______ proper subsets.
In the algebra class, it takes 479,001,600 days for a seating arrangement to repeat. There are 362,880 seating arrangements with Larry, Moe, and Curly in the front seats. Set S has 16 subsets, including 14 proper subsets.
To determine how many days must pass before a seating arrangement is repeated in an algebra class with 12 students and 12 desks, we can use the concept of permutations.
Since each student can sit in any of the 12 desks, the total number of possible seating arrangements is 12 factorial (12!). Therefore, the class can have 12! = 479,001,600 different seating arrangements.
To find out how many seating arrangements put Larry, Moe, and Curly in the front seats, we consider them as a group and arrange them in the front row.
The remaining 9 students can be seated in the back row, so the number of seating arrangements with Larry, Moe, and Curly in the front seats is 9 factorial (9!).
Therefore, there are 9! = 362,880 seating arrangements that put Larry, Moe, and Curly in the front seats.
For the set S = {y | y ∈ N and 103 ≤ y < 119}, we need to find the number of elements in this set. The range is from 103 to 118 (since 119 is not included), and the set contains natural numbers.
Therefore, the number of elements in this set is 118 - 103 + 1 = 16.
The set S has 16 subsets. This includes the empty set and the set itself, which are always subsets of any set.
Additionally, there are subsets containing a single element, subsets containing two elements, subsets containing three elements, and so on, up to subsets containing all 16 elements.
However, since we need to find proper subsets (excluding the empty set and the set itself), the number of proper subsets of set S is 16 - 2 = 14.
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