The value of $\sum\limits_{r = 0}^6 {\left( {{}^6{C_r}\,.\,{}^6{C_{6 - r}}} \right)}$ is equal to :
Solution
Given,<br><br>$\sum\limits_{r = 0}^6 {{}^6{C_r}{}^6{C_{6 - r}}}$
<br><br>= ${}^6{C_0}.{}^6{C_6} + {}^6{C_1}.{}^6{C_5} + ... + {}^6{C_6}.{}^6{C_0}$
<br><br>Now,
<br><br>$$\eqalign{
& = \left( {{}^6{C_0} + {}^6{C_1}x + {}^6{C_2}{x^2} + ... + {}^6{C_6}{x^6}} \right) \cr
& \left( {{}^6{C_0} + {}^6{C_1}x + {}^6{C_2}{x^2} + ... + {}^6{C_6}{x^6}} \right) \cr} $$
<br><br>Comparing coefficient of x<sup>6</sup> both sides
<br><br>${}^6{C_0}.{}^6{C_6} + {}^6{C_1}.{}^6{C_5} + ... + {}^6{C_6}.{}^6{C_0}$
<br><br>= ${}^{12}{C_6}$ = 924
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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