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Question

The value of n1r=0nCrnCr+nCr+1 is equals to ..............

A

n + 1

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B

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C

n + 2

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D

n

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Solution

The correct option is B


Let's try to change this expression into a general expression so that we can simplify it.
We know the general expression nCr+1nCr=nrr+1

So, now we will divide numerator and denominator by nCr to get a generalized expression.
n1r=0nCrnCr+nCr+1=n1r=011+nCr+1Cr=n1r=011+nrr+1=n1r=01r+1+nrr+1

=n1r=0r+1n+1=1n+1n1r=0(r+1)
=1(n+1)[1+2+.....+n]
But we know, Sum of the first of the first n terms = n(n+1)2
Therefore,
=n(n+1)(n+1)2=n2

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