Permutation and Combination

(11) A pizza menu offers 5 different toppings. How many different pizzas can be created by choosing 3 of these toppings?

• (a) 15
• (b) 10
• (c) 20
• (d) 25

Answer

Correct Answer: (b) 10
The number of different pizzas that can be created by choosing 3 toppings out of 5 is ^{5}C_{3} = \frac{5!}{3!(5-3)!} = 10 .

(12) In how many ways can you arrange the letters of the word “BANANA”?

• (a) 120
• (b) 60
• (c) 20
• (d) 720

Answer

Correct Answer: (b) 60
The number of ways to arrange the letters of the word “BANANA” is ^{6}P_{6} = 6! = 720 , but since there are repeating letters, we need to divide by the factorial of the number of times each letter repeats, which is 3! \times 2! = 12 . So, the total arrangements are \frac{720}{12} = 60 .

(13) How many different ways can you arrange the letters of the word “BOOK”?

• (a) 24
• (b) 12
• (c) 24
• (d) 6

Answer

Correct Answer: (c) 24
The number of different ways to arrange the letters of the word “BOOK” is \frac{4!}{2!} = 24 , where we divide by 2! to account for the repeated letter ‘O.’

(14) In how many ways can 3 red balls and 2 blue balls be arranged in a row?

• (a) 10
• (b) 15
• (c) 20
• (d) 12

Answer

Correct Answer: (c) 20
The number of ways to arrange 3 red balls and 2 blue balls in a row is \frac{5!}{3! \times 2!} = 20 .

(15) A box contains 5 red balls, 3 green balls, and 2 blue balls. How many different ways can you choose 4 balls from the box?

• (a) 90
• (b) 126
• (c) 84
• (d) 45

Answer

Correct Answer: (b) 126
The number of different ways to choose 4 balls from the box containing 5 red balls, 3 green balls, and 2 blue balls is ^{10}C_{4} = \frac{10!}{4!(10-4)!} = 126 .