Romi has a collection of 10 distinct books (8 small and 2 large). In how many ways can he select 5 books to take with him on a trip if he has room for only 1 large book?
A) 56
B) 126
C) 152
D) 196
E) 252
Answer: D
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Romi has a collection of 10 distinct books (8 small and 2 large). In how many ways can he select 5 books to take with
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Romi has a collection of 10 distinct books (8 small and 2 large). In how many ways can he select 5 books to take with him on a trip if he has room for only 1 large book?
Romi has two options available.
Either select 5 small books or select 4 small and 1 large book
Number of ways in which 5 small book can be selected = 8C5 = 56 ways
Number of ways in which 4 small and 1 large book can be selected = 8C4 * 2C1 = 70 * 2 = 140 ways
Total number of ways = 56 + 140 = 196 ways
Option D is the answer.
Romi has two options available.
Either select 5 small books or select 4 small and 1 large book
Number of ways in which 5 small book can be selected = 8C5 = 56 ways
Number of ways in which 4 small and 1 large book can be selected = 8C4 * 2C1 = 70 * 2 = 140 ways
Total number of ways = 56 + 140 = 196 ways
Option D is the answer.
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Solution:
The number of ways he can take exactly 1 large book (along with 4 small books) is to choose 1 large book out of 2 available and 4 small books out of 8 available. Thus, we have: 2C1 x 8C4 = 2 x (8 x 7 x 6 x 5)/(4 x 3 x 2) = 2 x 7 x 2 x 5 = 140 ways.
The number of ways he can take no large book (and thus all 5 small books) is 2C0 x 8C5 = 1 x (8 x 7 x 6 x 5 x 4)/(5 x 4 x 3 x 2) = 8 x 7 = 56.
Therefore, the total number of ways he can take at most 1 large book with him on the trip is 140 + 56 = 196.
Answer: D
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