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Question

Prove that nCr+n1Cr+n2Cr+.....+rCr=n+1Cr+1.

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Solution

To prove:
nCr+n1Cr+n2Cr+......+rCr=n+1Cr+1
nCr+n1Cr+n2Cr+.....+r+1Cr+rCr=n+1Cr+1
[nCn=n+1Cn+1]
nCr+n1Cr+n2Cr+......+r+2Cr+(r+1Cr+r+1Cr+1)
r+2Cr+1
[We know that n+1Cr+1=nCr+1+nCr]
nCr+n1Cr+n2Cr+......+r+2Cr+r+2Cr+1=n+1Cr+1
nCr+n1Cr+n2Cr+......+r+3Cr+1=n+1Cr+1
By continuing this, we get
nCr+(n1Cr+n1Cr+1)=n+1Cr+1
nCr+nCr+1=n+1Cr+1
n+1Cr+1=n+1Cr+1
L.H.S=R.H.S
[Hence proved].

1178549_827283_ans_049f7880d91449c8a4e481e17a2d62e3.jpg

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