Question

Prove from combination formula C(n,0)=n! For r=0

Answer 1

`C(n,r) = (n!)/(r!*(n-r)!)\\\\C(n,0) = (n!)/(0!*(n-0)!)\\\\C(n,0) = (n!)/(1*n!)\\\\C(n,0) = (n!)/(n!)\\\\C(n,0) = 1\\\\`

Your teacher made a mistake. The result should be 1 and not the factorial n! since those terms will cancel.

If you had n people, and you choose none of them, then there's only one way to do so. It's equivalent to the number of ways of selecting all n people where order doesn't matter.

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If your teacher was thinking about the permutation formula, then,

`P(n,r) = (n!)/((n-r)!)\\\\P(n,0) = (n!)/((n-0)!)\\\\P(n,0) = (n!)/(n!)\\\\P(n,0) = 1\\\\`

We get the same situation.

However, if r = n, we'll have:

`P(n,r) = (n!)/((n-r)!)\\\\P(n,n) = (n!)/((n-n)!)\\\\P(n,n) = (n!)/(0!)\\\\P(n,n) = (n!)/(1)\\\\P(n,n) = n!\\\\`

And finally we arrive at the factorial n! when using the permutation formula and when r = n.