Question

Find 9P9 and 9C9. Why do the answers differ?​

Answer 1

We have been given two expressions $9P9\text{ and }9C9$. We are asked to find the value of each.

To find 9P9, we will use permutations formula.

$^nP_r=\frac{n!}{(n-r)!}$, where

P = Number of permutations,

n = The total number of objects in the set,

r = Number of objects being chosen from the set.

$9P9=\frac{9!}{(9-9)!}$

$9P9=\frac{9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{(0)!}$

$9P9=\frac{362880}{1}$                   Using $0!=1$

$9P9=362880$

To find 9C9, we will use combinations formula.

$^nC_r=\frac{n!}{r!(n-r)!}$, where

C = Number of combinations,

n = The total number of objects in the set,

r = Number of objects being chosen from the set.

$9C9=\frac{9!}{9!(9-9)!}$

$9C9=\frac{9!}{9!(0)!}$

$9C9=\frac{9!}{9!\cdot 1}$     Using $0!=1$

Cancelling out $9!$, we will get:

$9C9=\frac{1}{1}$

$9C9=1$

The answers differ because order. With permutations we care about the order of the elements, while with combinations we don't.