Let n,k=^nC_k for 0 k n, else 0.

Question

Let $\langle n,k\rangle={}^nC_k$ for $0\le k\le n$, else $0$. If $A_k=\displaystyle\sum_{i=0}^{9}\langle9,i\rangle\langle12,12-k+i\rangle+\sum_{i=0}^{8}\langle8,i\rangle\langle13,13-k+i\rangle$ and $A_4-A_3=190p$, then $p$ is equal to

Answer: 49

Accepted Answer

By Vandermonde's identity, $A_k = {}^{21}C_k+{}^{21}C_k=2\cdot{}^{21}C_k$ (since $9+12=21$ and $8+13=21$).

$$A_4-A_3=2\left({}^{21}C_4-{}^{21}C_3\right)=2(5985-1330)=2\times4655=9310$$

$$190p=9310\quad\Rightarrow\quad p=49$$

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