Let $\left( {\matrix{
n \cr
k \cr
} } \right)$ denotes ${}^n{C_k}$ and $$\left[ {\matrix{
n \cr
k \cr
} } \right] = \left\{ {\matrix{
{\left( {\matrix{
n \cr
k \cr
} } \right),} & {if\,0 \le k \le n} \cr
{0,} & {otherwise} \cr
} } \right.$$
If $${A_k} = \sum\limits_{i = 0}^9 {\left( {\matrix{
9 \cr
i \cr
} } \right)\left[ {\matrix{
{12} \cr
{12 - k + i} \cr
} } \right] + } \sum\limits_{i = 0}^8 {\left( {\matrix{
8 \cr
i \cr
} } \right)\left[ {\matrix{
{13} \cr
{13 - k + i} \cr
} } \right]} $$ and A4 $-$ A3 = 190 p, then p is equal to :
Answer (integer)
49
Solution
$${A_k} = \sum\limits_{i = 0}^9 {{}^9{C_i}} {}^{12}{C_{k - i}} + \sum\limits_{i = 0}^8 {{}^8{C_i}} {}^{13}{C_{k - i}}$$<br><br>${A_k} = {}^{21}{C_k} + {}^{21}{C_k} = 2.{}^{21}{C_k}$<br><br>${A_4} - {A_3} = 2\left( {{}^{21}{C_4} - {}^{21}{C_3}} \right) = 2(5985 - 1330)$<br><br>$190p = 2(5985 - 1330) \Rightarrow p = 49$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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