Question

part one - multiple choice

1. the library labels each book with a five - digit code to identify the book in their computer system. each code is 3 digits (0 - 9) followed by two letters. how many codes are possible?

a. 468 c. 527,000
b. 492,804 d. 676,000

1. a test has 30 multiple choice questions with 4 choices each. which expressions represents the total number of ways to answer the questions?

a. ( 4^{30} ) c. ( _{30}p_4 )
b. ( 30^{4} ) d. ( 4cdot30 )

1. a coach needs to choose two players from the hockey team to be captain and assistant captain. if there are 20 players on the team, how many ways can he assign the two roles?

a. 100 c. 380
b. 190 d. 450

1. in a card game, each player is dealt a hand of 7 cards from a deck of 52 playing cards. which expression represents the number of hands possible?

a. ( 7!cdot52! ) c. ( _{52}p_7 )
b. ( \frac{52!}{7!} ) d. ( _{52}c_7 )

1. how many ways can the 11 letters in the word probability be arranged?

a. 1,663,200 c. 9,979,200
b. 7,241,600 d. 39,916,800

Explanation:

Question 1

Step1: Analyze digit choices

Each of the 3 digits has 10 options (0 - 9), so total for digits: \(10\times10\times10 = 10^3\)

Step2: Analyze letter choices

Each of the 2 letters has 26 options (A - Z), so total for letters: \(26\times26=26^2\)

Step3: Total codes

Multiply digit and letter possibilities: \(10^3\times26^2 = 1000\times676 = 676000\)

Each of the 30 questions has 4 choices. For each question, 4 options, so for 30 questions, it's \(4\times4\times\cdots\times4\) (30 times), which is \(4^{30}\)

Step1: Permutation formula

We need to choose 2 players out of 20 and assign different roles (captain and assistant), so use permutation \(_{n}P_{r}=\frac{n!}{(n - r)!}\), here \(n = 20\), \(r = 2\)

Step2: Calculate permutation

\(_{20}P_{2}=\frac{20!}{(20 - 2)!}=\frac{20!}{18!}=20\times19 = 380\)

Answer:

D. 676,000

Question 2