Make sure to use induction to prove your result. Recall the following concerning

Question

Make sure to use induction to prove your result.

Recall the following concerning Pochhammer's symbol: nk[Pochhammer's symbol] Given: Use the principle of mathematical induction to Prove that is true for all non-negative integers n.

Explanation / Answer

we need to prove n(n-1)(n-2).....(n-(k-1)) = n!/(n-k)! step 1: checking for k=1. for n=1, L.H.S = n. R.H.S = n!/(n-1)! = [n*(n-1)!]/(n-1)! = n hence L.H.S = R.H.S step 2: let the equation be true for k=m. m>1 i.e. n(n-1)(n-2).....(n-(m-1)) = n!/(n-m)! step 3: we prove for k=m+1 L.H.S = n(n-1)(n-2).....(n-(m-1))(n-m) = [n!/(n-m)!]*(n-m) denominator can be written as (n-m)! = (n-m)*(n-(m+1))! so L.H.S = {n!/[(n-m)*(n-(m+1)!)]}*(n-m) = n!/(n-(m+1))! R.H.S = n!/(n-(m+1))! so L.H.S = R.H.S hence we proved that given equation is true for k=m implies it is true for k=m+1. hence by the principle of mathematical induction, n(n-1)(n-2).....(n-(k-1)) = n!/(n-k)! for all nonzero k.

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