Question

If \((2 \leq r \leq n)\), then \({ }^{n} C_{r}+2 .{ }^{n} C_{r+1}+{ }^{n} C_{r+2}\) is equal to
(A) 2. \({ }^{n} \mathrm{C}_{\mathrm{r}+2}\)
(B) \({ }^{n+1} C_{r+1}\)
(C) \({ }^{n+2} \mathrm{C}_{r+2}\)
(D) \({ }^{n+1} C_{r}\)

Answer 1

Ans: (C)
Hint \(:{ }^{n} C_{r}+2{ }^{n} C_{r+1}+{ }^{n} C_{r+2}\)
$$
={ }^{n} C_{r}+{ }^{n} C_{r+1}+{ }^{n} C_{r+1}+{ }^{n} C_{r+2}={ }^{n+1} C_{r+1}+{ }^{n+1} C_{r+2}={ }^{n+2} C_{r+2}
$$