Question 1044456: Prove that nCr/nCr+nCr+1 = r+1/n+1
Please help me with this as soon as possible
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! nCr=n!/r!(n-r)!
nC(r+1)=n!/(r+1)!(n-r-1)!, the sign in this is different from the prior problem. Use numbers to show why, for example 7C3.
The denominator is
n!/r!(n-r)!+n!/(r+1)!(n-r-1)!
This has a common denominator of (r+1)!(n-r)!
Using that common denominator, the numerator is n!(r+1)+n!(n-r), because r! divides into (r+1)! r+1 times and n-r-1 divides into (n-r) n-r times.
The numerator of the whole thing is
n!/r!(n-r)!
The denominator is n!(r+1)+n!(n-r) all divided by (r+1)!(n-r)!
Every n! cancels because there is one in each term.
you now have
1/r!(n-r)! divided by {(r+1)+(n-r)]/(r+1)!(n-r)!}
the (n-r)! cancel and the fraction is (n+1)/(r+1)!
You now have
1/r! divided by (n+1)/(r+1)!
Invert and this is (r+1)!/r!(n+1)
but (r+1)! is (r+1)*r!, and the r! cancel, so you have
(r+1)/(n+1)
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