Question
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
A.
$$^{n + 1}{C_{r + 1}}$$
B.
$$^{n + 2}{C_r}$$
C.
$$^{n + 2}{C_{r + 1}}$$
D.
$$^{n + 1}{C_r}$$
Answer :
$$^{n + 2}{C_{r + 1}}$$
Solution :
$$\eqalign{
& ^n{C_{r + 1}} + {\,^n}{C_{r - 1}} + 2{\,^n}{C_r} = {\,^n}{C_{r - 1}} + {\,^n}{C_r} + {\,^n}{C_r} + {\,^n}{C_{r + 1}} \cr
& = {\,^{n + 1}}{C_r} + {\,^{n + 1}}{C_{r + 1}} = {\,^{n + 2}}{C_{r + 1}} \cr} $$