Question

show nPn = n! & nCn = 1​

Answer 1

Given:

Permutation and Combination​

To find:

Show nPn = n! & nCn = 1​

Solution:

Permutation

nPn is “how many ways can we arrange n in a row?”  

we use the formula,

nPr = n!/(n - r)!

taking r = n, we get,  

nPn = n!/(n - n)! = n!/0! = n!/1 = n!                     [factorial 0! = 1]

∴ nPn = n!  hence the proof.

Combination

nCn is “how many ways can you choose n from a pile of n.”  

we use the formula,

nCr = n!/[r! (n - r)!]

taking r = n, we get,

nCn = n!/[n! (n - n)!] = 1/0! = 1/1 = 1

∴ nCn = 1 hence the proof.