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

Aug 10, 2017

$1140$ 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..

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)

$\Rightarrow \textcolor{w h i t e}{x} \frac{20 \times 19 \times 18 \times \cancel{17} \times \cancel{16} \times \cancel{15} \times \ldots . . \times \cancel{3} \times \cancel{2} \times \cancel{1}}{\left(\cancel{17} \times \cancel{16} \times \cancel{15} \times \ldots . . \times \cancel{3} \times \cancel{2} \times \cancel{1}\right) \times \left(3 \times 2 \times 1\right)}$

$\Rightarrow \textcolor{w h i t e}{x} \frac{20 \times 19 \times 18}{3 \times 2 \times 1}$

$\Rightarrow \textcolor{w h i t e}{x} \frac{6840}{6}$

$\Rightarrow \textcolor{w h i t e}{x} 1140$ ways

Hope that's clear??

Aug 10, 2017

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 $3 \times 2 \times 1 = 6$ ways of arranging three ties.

So the total number of possible combinations is

$\frac{20 \times 19 \times 18}{3 \times 2 \times 1} = \frac{6840}{6} = 1140$