Question

suppose we want to choose 5 colors, without replacement, from 18 distinct colors. (a) how many ways can this be done, if the order of the choices is taken into consideration? (b) how many ways can this be done, if the order of the choices is not taken into consideration?

Explanation:

Step1: Recall permutation formula

When order matters, we use the permutation formula $P(n,r)=\frac{n!}{(n - r)!}$, where $n = 18$ and $r=5$.
$P(18,5)=\frac{18!}{(18 - 5)!}=\frac{18!}{13!}=18\times17\times16\times15\times14$

Step2: Calculate the value

$18\times17\times16\times15\times14 = 1028160$

Step3: Recall combination formula

When order does not matter, we use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, with $n = 18$ and $r = 5$.
$C(18,5)=\frac{18!}{5!(18 - 5)!}=\frac{18!}{5!×13!}=\frac{18\times17\times16\times15\times14}{5\times4\times3\times2\times1}$

Step4: Calculate the value

$\frac{18\times17\times16\times15\times14}{5\times4\times3\times2\times1}=8568$

Answer:

(a) 1028160
(b) 8568