Q. If $\ ^{n}C_{r}$ denotes the number of combinations of $n$ things takes $r$ at a time, then the expression $\ ^{n}C_{r + 1} + \ ^{n}C_{r - 1} + 2 \times \ ^{n}C_{r}$, equals
A) $\ ^{n + 2}C_{r}$
B) $\ ^{n + 2}C_{r + 1}$
C) $\ ^{n + 1}C_{r}$
D) $\ ^{n + 1}C_{r + 1}$
Explanation
(b)
Now, $\ ^{n}C_{r + 1} + \ ^{n}C_{r - 1} + 2.\ ^{n}C_{r}$
$= \ ^{n}C_{r + 1} + \ ^{n}C_{r} + \ ^{n}C_{r - 1} + \ ^{n}C_{r}$
$= \ ^{n + 1}C_{r + 1} + \ ^{n + 1}C_{r} = \ ^{n + 2}C_{r + 1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.