Question

From $12$ books in how many ways can a selection of $5$ be made, (1) when one specified book is always included; (2) when on specified book is always excluded?

Answer 1

(i) Since the specified book is to be included in every selection, we have only to choose $4$ out of the remaining $11$

Hence the number of ways $=$ ${}^{11}{C}_4$

$=\frac{11\times 10\times 9\times 8}{1\times 2\times 3\times 4}=330$

(2) Since the specified book is always to be excluded, we have to select the $5$ books out of the remaining $11$.

Hence the number of ways $=$ ${}^{11}{C}_5$

$=\frac{11\times 10\times 9\times 8\times 7}{1\times 2\times 3\times 4\times 5}=462$