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Question

Mathematics Question on Combinations

mCr+1^mC_{r+1} + ∑k=mn\sum_{k =m}^{n} kCr=^kC_r =

A

nCr+1^nC_{r+1}

B

n+1Cr+1^{n+1} C_{r+1}

C

nCr^nC_r

D

None of these

Answer

n+1Cr+1^{n+1} C_{r+1}

Explanation

Solution

mCr+1^{m}C_{r+1}+ ∑k=mn\displaystyle\sum_{k=m}^{n} kCr^{k}C_{r} =(mCr+1+mCr)+m+1Cr+m+2Cr+…m+nCr=\left(^{m}C_{r+1}+^{m}C_{r}\right)+^{m+1}C_{r}+^{m+2}C_{r} +\ldots^{m+n}C_{r} =(m+1Cr+1+m+1Cr)+m+2Cr+⋯m+nCr=\left(^{m+1}C_{r+1}+^{m+1}C_{r}\right)+^{m+2}C_{r} +\cdots ^{m+n}C_{r} =m+2Cr+1+m+2Cr+…+nCr= ^{m+2}C_{r+1} +^{m+2}C_{r}+\ldots+ ^{n}C_{r} ::: :::\begin{matrix}:&:&:\\\ :&:&:\end{matrix} =n+1Cr+1=^{n+1}C_{r+1}