Question

From $12$books, the difference between the number of ways a selection of $5$ books when one specified book is always excluded and one specified book is always included, is equal to:

• A. $64$
• B. $118$
• C. $132$
• D. $330$
• E. $462$

Answer 1

The correct option is C

$132$

Explanation for correct option:

Calculate the required difference using combination.

The number of books can be chosen when one specified book is always excluded, is

$$\begin{gathered} C_4^{11} \\ =\frac{11!}{7!\times 4!} \\ =\frac{11\times 10\times 9\times 8\times 7!}{7!\times 4\times 3\times 2} \\ =330 \end{gathered}$$

Again, the number of books can be chosen when one specified book is always included, is

$$\begin{gathered} C_5^{11} \\ =\frac{11!}{6!\times 5!} \\ =\frac{11\times 10\times 9\times 8\times 7\times 6!}{6!\times 5\times 4\times 3\times 2} \\ =462 \end{gathered}$$

Therefore, $5$ books can be selected in ways when one specified book is always excluded and one specified book is always included, is:

$$\begin{gathered} =(462-330) \\ =132 \end{gathered}$$

Therefore, Option(C) is the correct answer.