Let nk denote ^nC_k and let: $[ matrix n \\ k matrix ] = cases nk, & if 0 k n \\ 0, & otherwise cases \\ If A_k = _i=0^9 9i [ matrix 12 \\ 12-k+i matrix ] + _i=0^8 8i [ matrix 13 \\ 13-k+i matrix ] \\ and A_4 - A_3 = 190p, then p is equal to \_\_\_\_

A_k = _i = 0^9 ^9C_i ^12C_k - i + _i = 0^8 ^8C_i ^13C_k - i\\ A_k = ^21C_k + ^21C_k = 2.^21C_k\\ A_4 - A_3 = 2( ^21C_4 - ^21C_3 ) = 2(5985 - 1330)\\ 190p =&n

Let n k denotes nCk and n k n k if0 le k le n 0 otherwi: (Mathematics)

Let

\binom{n}{k}

denote

^nC_k

and let:

\left[ \begin{matrix} n \\ k \end{matrix} \right] = \begin{cases} \binom{n}{k}, & \text{if } 0 \le k \le n \\ 0, & \text{otherwise} \end{cases} \\ \text{If } A_k = \sum_{i=0}^{9} \binom{9}{i} \left[ \begin{matrix} 12 \\ 12-k+i \end{matrix} \right] + \sum_{i=0}^{8} \binom{8}{i} \left[ \begin{matrix} 13 \\ 13-k+i \end{matrix} \right] \\ \text{and } A_4 - A_3 = 190p,

then p is equal to

\_\_\_\_\_\_\_\_\_

Explanation

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