Question

You have 20 different neckties in your wardrobe. How many combinations of three ties could you choose?

Answer 1

`1140` ways

Explanation:

From the detailed question i picked the word Combinations

Which i should believe the question is gotten from the topic; Permutation and Combination..

Follow this simple steps..

You have 20 Neck ties, out of 3 ties could you choose..

It goes with this formula of combination;

`"Combination Formula" rArr ^nC_r = (n!)/((n-r)! r!)`

Where `n = 20` and `r = 3`

`rArr (20!)/((20-3)! 3!)`

`rArr color(white)(x) (20!)/(17! 3!)`

`rArr color(white)(x) (20 xx 19 xx 18 xx 17 xx 16 xx 15 xx ........xx 3 xx 2 xx 1)/((17 xx 16 xx 15 xx ..... xx3 xx 2 xx 1)xx (3 xx 2 xx 1)`

`rArr color(white)(x) (20 xx 19 xx 18 xx cancel17 xx cancel16 xx cancel15 xx ..... xx cancel3 xx cancel2 xx cancel1)/((cancel17 xx cancel16 xx cancel15 xx ..... xx cancel3 xx cancel2 xx cancel1)xx (3 xx 2 xx 1))`

`rArr color(white)(x) (20 xx 19 xx 18)/(3 xx 2 xx 1)`

`rArr color(white)(x) 6840/6`

`rArr color(white)(x) 1140` ways

Hope that's clear??

Answer 2

There are `1140` different combinations if order is not important.

Explanation:

There will be:

`20` different choices for the first tie and then
`19` different choices for the second tie and then
`18` different choices for the third tie.

This gives `6840` possibilities

However within these the same groups will be repeated.

For example Red, Blue, Green, and Red, Green, Blue, and Blue,Red, Green are all the same combination of colours.

There are `3xx2xx1 =6` ways of arranging three ties.

So the total number of possible combinations is

`(20xx19xx18)/(3xx2xx1) = 6840/6 = 1140`