Problem Solving

Romi has a collection of 10 distinct books out of which 8 are small and 2 are large. In how many ways can he select 5 books to take with him on a trip if he can take at most 1 large book?

A 56
B 126
C 152
D 196
E 252

Total books = 10
Total small books = 8
Total large books = 2

In how many ways can Romi select 5 books to take with him on a trip if he can take at most 1 large book?

Since the maximum large book(s) he can take = 1
Therefore, he can select either - 5 small books or
- 4 small books and 1 large book

$$Selecting\ 5\ small\ book\ =\ 8C5$$
$$8C5\ =\frac{8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{5\cdot4\cdot3\cdot2\cdot1\left(3\cdot2\cdot1\right)}$$
$$=\frac{8\cdot7\cdot6}{6}$$
$$=8\cdot7=56$$

$$Selecting\ 4\ small\ books\ and\ 1\ l\arg e\ book\ =\ 8C4\cdot2C1$$
$$8C4=\frac{8!}{4!\left(8-4\right)!}$$
$$\frac{8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{4\cdot3\cdot2\cdot1\left(4\cdot3\cdot2\cdot1\right)}$$
$$\frac{8\cdot7\cdot6\cdot5}{4\cdot3\cdot2\cdot1}=\frac{1680}{24}$$
$$=70$$
$$2C1=\frac{2!}{1!\left(2-1\right)!}=\frac{2\cdot1}{1\cdot1}=2$$
$$8C4\ \cdot\ 2C1\ =\ 70\cdot2=140$$


The total number of ways for Romi to select 5 books = 56 +140 = 196
Answer = D