Question

The library is to be given 5 books as a gift. the books will be selected from a list of 21 titles. if each book selected must have a different title, how many possible selections are there?

Answer 1

20,349.

Explanation:

We have been that a library is to be given 5 books as a gift. the books will be selected from a list of 21 titles.

We will combinations to solve our given problem.

`_(r)^(n)\textrm{C}=(n!)/(r!(n-r)!)`

Different ways to choose 5 books from 21 tiles:

`_(5)^(21)\textrm{C}=(21!)/(5!(21-5)!)`

`_(5)^(21)\textrm{C}=(21!)/(5!(16)!)`

`_(5)^(21)\textrm{C}=(21*20*19*18*17*16!)/(5*4*3*2*1*(16)!)`

`_(5)^(21)\textrm{C}=(21*19*18*17)/(3*2)`

`_(5)^(21)\textrm{C}=(21*19*3*17)/(1)`

`_(5)^(21)\textrm{C}=20,349`

Therefore, there are 20349 possible selections.

Answer 2

This is a combination problem. Order does not matter and the books are all different. You would write this as 21C5 or
`\tbinom{21}{5}`. Combinations are not fractions.

The answer is
`(21 * 20 * 19 * 18 * 17)/(1 * 2 * 3 * 4 * 5)`

which = 20349 different ways these books can be chosen.