Q: The IATA assigns three-letter codes to airports using only uppercase English letters. How many different codes are possible?

$$26 \times 26 \times 26 = 17,576$$

Q: How many different simple random samples of size 4 can be obtained from a population of size 20?

$$\frac{20!}{4! \times 16!} = 4845$$

Q: How many different vertical arrangements are there of 10 flags if 5 are white, 3 are blue, and 2 are red?

$$\frac{10!}{5! \times 3! \times 2!} = 2520$$

Q: A password consists of 2 letters followed by 3 digits. How many different passwords are possible if repetition is allowed?

$$26^2$$ \times $$10^3 = 676$$,000$$

Q: A committee of 5 is to be formed from 8 men and 7 women. How many ways can the committee be formed if it must contain exactly 3 men and 2 women?

$$\binom{8}{3} \times \binom{7}{2} = 56 \times 21 = 1,176$$

Question 1

A restaurant offers 3 types of salads, 5 types of main courses, and 2 types of desserts. Using the multiplication rule of counting, how many different meal combinations are possible if you choose one of each?

Accepted Answer

$$3 	imes 5 	imes 2 = 30$$

Question 2

The IATA assigns three-letter codes to airports using only uppercase English letters. How many different codes are possible?

Accepted Answer

$$26 	imes 26 	imes 26 = 17,576$$

Question 3

A fair coin is tossed 10 times. What is the probability of observing at least one head?

Accepted Answer

$$1 - \frac{1}{1024} = 0.999$$

Question 4

How many ways can the top three finishers in a 10-horse race be arranged?

Accepted Answer

$$10 	imes 9 	imes 8 = 720$$

Question 5

How many different simple random samples of size 4 can be obtained from a population of size 20?

Accepted Answer

$$\frac{20!}{4! 	imes 16!} = 4845$$

Question 6

How many different vertical arrangements are there of 10 flags if 5 are white, 3 are blue, and 2 are red?

Accepted Answer

$$\frac{10!}{5! 	imes 3! 	imes 2!} = 2520$$

Question 7

In the Illinois Lottery, six balls are drawn from 52 without replacement and order does not matter. If a ticket has two sets of six numbers, what is the probability of winning?

Accepted Answer

$$\frac{2}{20,358,520} \approx 0.000000098$$

Question 8

Which formula gives the number of permutations of $$n$$ objects taken $$r at a $$time, where order matters and repetition is not allowed?

Accepted Answer

$$\frac{n!}{(n-r)!}$$

Question 9

Which formula gives the number of combinations of $$n$$ objects taken $$r at a $$time, where order does not matter and repetition is not allowed?

Accepted Answer

$$\frac{n!}{r!(n-r)!}$$

Question 10

A password consists of 2 letters followed by 3 digits. How many different passwords are possible if repetition is allowed?

Accepted Answer

$$26^2$$ 	imes $$10^3 = 676$$,000$$

Question 11

Suppose you have a population of 4 units labeled 1, 2, 3, 4. How many possible samples of size 2 are there under simple random sampling with replacement and order is important?

Accepted Answer

$$4^2 = 16$

Question 12

Suppose you have a population of 4 units labeled 1, 2, 3, 4. How many possible samples of size 2 are there under simple random sampling without replacement and order is not important?

Accepted Answer

$$\frac{4!}{2!2!} = 6$$

Question 13

A committee of 5 is to be formed from 8 men and 7 women. How many ways can the committee be formed if it must contain exactly 3 men and 2 women?

Accepted Answer

$$\binom{8}{3} 	imes \binom{7}{2} = 56 	imes 21 = 1,176$$

Question 14

How many ways can you arrange the letters in the word 'STATISTICS'?

Accepted Answer

$$\frac{10!}{3!3!2!2!}$$

Question 15

A student must answer 5 out of 10 questions on an exam. In how many ways can the student choose which questions to answer?

Accepted Answer

$$\binom{10}{5} = 252$$

Question 16

A traveling salesman must visit 4 warehouses, each in a different location, and return to the starting point. How many possible routes are there?

Accepted Answer

$$4! = 24$$

Question 17

Which of the following best describes the difference between a permutation and a combination?

Accepted Answer

A permutation considers order, while a combination does not.

Question 18

If you select 3 books from a shelf of 8 different books, and the order in which you select them matters, how many ways can you select the books?

Accepted Answer

$$\frac{8!}{(8-3)!} = 336$$

Counting Principles in Statistics: Permutations, Combinations, and Random Sampling

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