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  4. If nCr denotes the number of combination of n things taken r at a time, then the expression nCr+1 + nCr-1 + 2 × nCr equals

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Q. If $^nC_r$ denotes the number of combination of n things taken r at a time, then the expression $^nC_{r+1} + {^{n}C_{r-1}} + 2 \times {^{n}C_{r}}$ equals

Permutations and Combinations

Solution:

$^{n}C_{r+1} + ^{n}C_{r-1} + 2 \times \,{}^{n}C_{r} $
$= ^{n}C_{r-1} + ^{n}C_{r} + ^{n}C_{r} + ^{n}C_{r+1} $
$=^{n+1}C_{r} + ^{n+1}C_{r+1} = ^{n+2}C_{r+1} $