SOLUTION: Show that nPr = nPr+1 (show that n permutation r equals to n permutation r+1).

Question 1014498: Show that nPr = nPr+1 (show that n permutation r equals to n permutation r+1).
Answer by mathmate(429) (Show Source): You can put this solution on YOUR website!

Question:
Show that nPr = nPr+1 (show that n permutation r equals to n permutation r+1)

Solution:
We will use the notation , so that
if and when
P(n,r)=P(n,r+1), then
P(n,r)-P(n,r+1)=0.............(1)
expanding above
=0
=0
Add by cross multiplication:
=0.....(1a)
(1a) can be satisfied if and only if the numerator equals zero.
=>
n=0, trivial solution if r=0.
n=r, leads to (-1)! in denominator, rejected
n-r-1=0, means n=r+1
Thus
The above equation can be satisfied when n=r+1, or
P(n,n-1)=P(n,n) for all n>0.

Note: if there is a typo in the original question, please post a new question.

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