Question

Suppose you have 18 objects (10 of type A, 5 of type B, and 3 of type C). Objects of type A are indistinguishable from each other; objects of type B are indistinguishable from each other; and objects of type C are indistinguishable from each other. In how many ways can you Pick 5 of the 18 objects (order does not matter)

Answer 1

Given:

Number of objects: n = 18

Type A objects: 10

Type B objects: 5

Type C objects: 3

To find:

In how many ways can you Pick 5 of the 18 objects (order does not matter)

Explanation:

When the order does not matter we use Combination.

Formula to calculate combination:

C(n,r) = n! / r! ( n - r )!

n = 18

r = 5

Putting the values:

C(n,r)

= C(18,5)

= 18! / 5! ( 18 - 5 )!

= 18! / 5! ( 13 )!

= ( 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 ) / ( 5 * 4 * 3 * 2 * 1 ) * (13 * 12 * 11 * 10 *9* 8 * 7 *6 * 5 * 4 * 3 * 2 *1 )

Cancel 13!

= (18 * 17 * 16 * 15 * 14 ) / ( 5 * 4 * 3 * 2 * 1 )

= 1028160 / 120

= 8568

So you can pick 5 of the 18 objects in 8568 ways.

Answer 2

`\binom{18}{5}= 8568`

Explanation:

Note that we have in total 18 items. Even though we are given information regarding the amounts of items per type, the general question asks the total number of ways in which you can pick 5 out of the 18 objects, without any restriction on the type of chosen items. Therefore, the information regarding the type is unnecessary to solve the problem.

Recall that given n elements, the different ways of choosing k elements out of n is given by the binomial coefficient
`\binom{n}{k})`.

Therefore, in this case the total number of ways is just
`\binom{18}{5}=8568`