Question 1

Evaluate the expression.
-9 !

Accepted Answer

The answer of Evaluate the expression.
-9 !
...

Question 2

Evaluate the expression.
-$5^{\mathrm{P}} 4$

Accepted Answer

The answer of Evaluate the expression.
-$5^{\mathrm{P}} 4$
...

Question 3

Evaluate the expression.
-$8^{\mathrm{P}_{4}}$

Accepted Answer

The answer of Evaluate the expression.
-$8^{\mathrm{P}_{4}}$
...

Question 4

Evaluate the expression.
-$10^{P_{2}}$

Accepted Answer

The answer of Evaluate the expression.
-$10^{P_{2}}$
...

Question 5

Evaluate the expression.
-${ }_{6} \mathrm{P}_{5}$

Accepted Answer

The answer of Evaluate the expression.
-${ }_{6} \mathrm{P}_{5}$
...

Question 6

Evaluate the expression.
-${ }_{12} \mathrm{C}_{3}$

Accepted Answer

The answer of Evaluate the expression.
-${ }_{12} \mathrm{C}_{3}$
...

Question 7

Evaluate the expression.
-${ }_{12} C_{0}$

Accepted Answer

The expression ${ }_{12} C_{0}$ represents the number of ways to choose 0 items from 12, which is always 1.

Question 8

Evaluate the expression.
-${ }_{38} \mathrm{C}_{1}$

Accepted Answer

The expression ${ }_{38} \mathrm{C}_{1}$ represents the number of ways to choose 1 item from 38, which is simply 38.

Question 9

Evaluate the expression.
-${ }_{23} \mathrm{C}_{14}$

Accepted Answer

The answer of Evaluate the expression.
-${ }_{23} \mathrm{C}_{14}$
...

Question 10

Use the multiplication principle to solve the problem.
-How many different 5 -digit sequences can be formed using the digits $0,1, \ldots, 9$ if repetition of digits is allowed?

Accepted Answer

The answer of Use the multiplication principle to solve the...

Question 11

Use the multiplication principle to solve the problem.
-License plates are made using 3 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

Accepted Answer

There are 26 choices for each of the 3 letters and 10 choices for each of the 2 digits. Thus, the total number of plates is $26^3 \times 10^2 = 1,757,600$.

Question 12

Use the multiplication principle to solve the problem.
-If 11 newborn babies are randomly selected, how many different gender sequences are possible?

Accepted Answer

Each baby can be either male or female, giving 2 choices per baby. For 11 babies, the number of sequences is $2^{11} = 2048$.

Question 13

Use the multiplication principle to solve the problem.
-A restaurant offers 6 possible appetizers, 10 possible main courses, and 4 possible desserts. How many different meals are possible at this restaurant?
(Two meals are considered different unless all three courses are the same).

Accepted Answer

The multiplication principle states that if there are multiple choices to be made, the total number of combinations is the product of the number of choices for each decision. Here, there are 6 choices for appetizers, 10 choices for main courses, and 4 choices for desserts. Therefore, the total number of different meals is 6 * 10 * 4 = 240.

Question 14

Use the multiplication principle to solve the problem.
-How many different 4 -letter radio station call letters can be made if repeats are allowed and the first letter must be $\mathrm{K}$.

Accepted Answer

The first letter is fixed as 'K', leaving 3 positions to be filled with any of the 26 letters of the alphabet. Since repeats are allowed, each of these 3 positions can be filled in 26 ways. Thus, the total number of combinations is $26^3 = 17,576$.

Question 15

Use the multiplication principle to solve the problem.
-How many different 7 -digit phone numbers are possible if the first digit cannot be a 0 ?

Accepted Answer

The first digit has 9 possible choices (1-9), and each of the remaining 6 digits has 10 possible choices (0-9). Therefore, the total number of different phone numbers is $9 \times 10^6 = 9,000,000$.

Question 16

Use the multiplication principle to solve the problem.
-A biologist is attempting to classify plant species by assigning 6 initials to each species. How many classifications are possible using this method?

Accepted Answer

Each initial can be one of 26 letters (A-Z). Therefore, the number of possible classifications is $26^6$, which equals 308,915,776.

Question 17

Use the multiplication principle to solve the problem.
-Mark can remember only the first 5 digits of his friend's phone number. He also knows that the number has 7 digits and that the last digit is not a 0 . If Mark were to dial all of the possible numbers and if it takes him 16 seconds to try each one, how long would it take to try every possibility?

Accepted Answer

The answer of Use the multiplication principle to solve the...

Question 18

Solve the problem.
-A musician plans to perform 10 selections. In how many ways can she arrange the musical selections?

Accepted Answer

The number of ways to arrange 10 selections is given by the factorial of 10, which is $10! = 3,628,800$.

Question 19

Solve the problem.
-A pollster wants to minimize the effect the order of the questions has on a person's response to a survey. How many different surveys are required to cover all possible arrangements if there are 8 questions on the survey?

Accepted Answer

The number of different arrangements of 8 questions is given by the factorial of 8, which is 8! = 40,320.

Question 20

Solve the problem.
-There are 10 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible?

Accepted Answer

The number of ways to elect a chairperson, a secretary, and a treasurer from 10 members is a permutation problem. The formula for permutations is $ nPr = \frac{n!}{(n-r)!} $. Here, $ n = 10 $ and $ r = 3 $. So, $ 10P3 = \frac{10!}{(10-3)!} = \frac{10 \times 9 \times 8}{1} = 720 $.

Quiz 9: Counting, Probability Distributions, and Further Topics in Probability

Evaluate the expression. -9 !

Evaluate the expression. - $5^{\mathrm{P}} 4$

Evaluate the expression. - $8^{\mathrm{P}_{4}}$

Evaluate the expression. - $10^{P_{2}}$

Evaluate the expression. - ${ }_{6} \mathrm{P}_{5}$

Evaluate the expression. - ${ }_{12} \mathrm{C}_{3}$

Evaluate the expression. - ${ }_{12} C_{0}$

Evaluate the expression. - ${ }_{38} \mathrm{C}_{1}$

Evaluate the expression. - ${ }_{23} \mathrm{C}_{14}$

Use the multiplication principle to solve the problem. -How many different 5 -digit sequences can be formed using the digits $0,1, \ldots, 9$ if repetition of digits is allowed?

Use the multiplication principle to solve the problem. -License plates are made using 3 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

Use the multiplication principle to solve the problem. -If 11 newborn babies are randomly selected, how many different gender sequences are possible?

Use the multiplication principle to solve the problem. -A restaurant offers 6 possible appetizers, 10 possible main courses, and 4 possible desserts. How many different meals are possible at this restaurant? (Two meals are considered different unless all three courses are the same) .

Use the multiplication principle to solve the problem. -How many different 4 -letter radio station call letters can be made if repeats are allowed and the first letter must be $\mathrm{K}$ .

Use the multiplication principle to solve the problem. -How many different 7 -digit phone numbers are possible if the first digit cannot be a 0 ?

Use the multiplication principle to solve the problem. -A biologist is attempting to classify plant species by assigning 6 initials to each species. How many classifications are possible using this method?

Solve the problem. -A musician plans to perform 10 selections. In how many ways can she arrange the musical selections?

Solve the problem. -A pollster wants to minimize the effect the order of the questions has on a person's response to a survey. How many different surveys are required to cover all possible arrangements if there are 8 questions on the survey?

Solve the problem. -There are 10 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible?

Quiz 9: Counting, Probability Distributions, and Further Topics in Probability

Evaluate the expression. -9 !

Evaluate the expression. - 5P45^{\mathrm{P}} 45P4

Evaluate the expression. - 8P48^{\mathrm{P}_{4}}8P4​

Evaluate the expression. - 10P210^{P_{2}}10P2​

Evaluate the expression. - 6P5{ }_{6} \mathrm{P}_{5}6​P5​

Evaluate the expression. - 12C3{ }_{12} \mathrm{C}_{3}12​C3​

Evaluate the expression. - 12C0{ }_{12} C_{0}12​C0​

Evaluate the expression. - 38C1{ }_{38} \mathrm{C}_{1}38​C1​

Evaluate the expression. - 23C14{ }_{23} \mathrm{C}_{14}23​C14​

Use the multiplication principle to solve the problem. -How many different 5 -digit sequences can be formed using the digits 0,1,…,90,1, \ldots, 90,1,…,9 if repetition of digits is allowed?