jee-main 2022 Q63

jee-main · India · session1_28jun_shift1 Combinations & Selection Combinatorial Identity or Bijection Proof
If $\sum_{k=1}^{31} \left({}^{31}\mathrm{C}_k\right)\left({}^{31}\mathrm{C}_{k-1}\right) - \sum_{k=1}^{30}\left({}^{30}\mathrm{C}_k\right)\left({}^{30}\mathrm{C}_{k-1}\right) = \frac{\alpha(60!)}{(30!)(31!)}$, where $\alpha \in R$, then the value of $16\alpha$ is equal to
(1) 1411
(2) 1320
(3) 1615
(4) 1855
If $\sum_{k=1}^{31} \left({}^{31}\mathrm{C}_k\right)\left({}^{31}\mathrm{C}_{k-1}\right) - \sum_{k=1}^{30}\left({}^{30}\mathrm{C}_k\right)\left({}^{30}\mathrm{C}_{k-1}\right) = \frac{\alpha(60!)}{(30!)(31!)}$, where $\alpha \in R$, then the value of $16\alpha$ is equal to\\
(1) 1411\\
(2) 1320\\
(3) 1615\\
(4) 1855