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

Question

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 (A) n +1Cr + 1 (B) n +2Cr  (C) n +2Cr + 1 (D) n +1Cr

Tardigrade Admin · Accepted Answer

nCr+1 + nCr-1 + 2 × nCr = nCr-1 + nCr + nCr + nCr+1 =n+1Cr + n+1Cr+1 = n+2Cr+1

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

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

$^{n + 2} C_{r}$

$^{n + 2} C_{r + 1}$

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

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

Q. 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

n+1Cr+1​

n+2Cr​

n+2Cr+1​

n+1Cr​