Verify that

`^nC_(r)` = `(n(n-1)(n-2)...(n-r+1))/(r!)`

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`^nC_r` is n taken r at a time. The expression for this is:

`^nC_r = (n!)/((n-r)!(r!))`

Note that `n! = n*(n-1)*(n-2)*(n-3)*(n-4)* cdots * 2 * 1` .

Note also that the expression `^nC_r` is only defined if `r <= n` . Hence, we can write n! as:

`n! = n* (n-1)*(n-2) * cdots * (n - r + 1)* (n - r) * (n - r - 1) * cdots * 2 * 1`

for some r less than or equal to n.

Note also that `(n-r)! = (n-r)*(n-r-1)*(n-r-2)* cdots * 2 * 1` .

Hence:

`^nC_r = (n*(n-1)*(n-2)* cdots * (n-r+1)*(n-r)*(n-r-1)* cdots * 2*1)/(((n-r)*(n-r-1)*(n-r-2)* cdots * 2 * 1)*(r!))`

`^nC_r = (n*(n-1)*(n-2) * cdots * (n-r + 1))/(r!)` , which is what we want to show.

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