Q3 of 4 Page 138

Evaluate: nCr + n-1Cr + n-2Cr + n-3Cr + … + r+1Cr + rCr

Above question boils down to:


Sn = nCr + n-1Cr + n-2Cr + n-3Cr + … + r+1Cr + rCr




We will Prove this by mathematical induction:


Sr = rcr = 1; RHS = r+1Cr+1 = 1


Sr+1 = rcr + r+1cr = 1 + r+1 = r + 2


Let Sk = k+1cr+1


Sk+1 = k+1cr+1 + k+1cr







= k+2Cr+1


= RHS


So, nCr + n-1Cr + n-2Cr + n-3Cr + …+ rCr = n+1Cr+1


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