Q65 / 155 · WBJEE 2018 · Mathematics
$$
\text { If }(2 \leq r \leq n), \text { then }{ }^{n} C_{r}+2 \cdot{ }^{n} C_{r+1}+{ }^{n} C_{r+2} \text { is }
$$
equal to
- A$2 \cdot{ }^{n} C_{r+7}$
- B$"^{-1} C_{r+1}$
- Correct answerC${ }^{n+2} C_{r+2}$
- D$^{n+1} C_{r}$
Solution
${ }^{n} C_{r}+2 \cdot{ }^{n} C_{r+1}+{ }^{n} C_{r+2}$
$={ }^{n} C_{r}+{ }^{n} C_{r+1}+{ }^{n} C_{r+1}+{ }^{n} C_{r+2}$
$={ }^{n+1} C_{r+1}+{ }^{n+1} C_{r+2}$
$\quad\left(\because{ }^{n} C_{r}+{ }^{n} C_{r+1}={ }^{n+1} C_{r+1}\right)$
$={ }^{n+2} C_{r+2}$
Hence, the correct answer is option: 'C'.