Question 318 of 347MediumMCQ
The value of \[\sum\limits_{r = 0}^6 {\left( {{}^6{C_r}\,.\,{}^6{C_{6 - r}}} \right)} \] is equal to :
Solutions · 1
Official solution
Given,
\[\sum\limits_{r = 0}^6 {{}^6{C_r}{}^6{C_{6 - r}}} \]
= \[{}^6{C_0}.{}^6{C_6} + {}^6{C_1}.{}^6{C_5} + ... + {}^6{C_6}.{}^6{C_0}\]
Now,
\[\eqalign{
\& = \left( {{}^6{C_0} + {}^6{C_1}x + {}^6{C_2}{x^2} + ... + {}^6{C_6}{x^6}} \right) \cr
\& \left( {{}^6{C_0} + {}^6{C_1}x + {}^6{C_2}{x^2} + ... + {}^6{C_6}{x^6}} \right) \cr} \]
Comparing coefficient of x6 both sides
\[{}^6{C_0}.{}^6{C_6} + {}^6{C_1}.{}^6{C_5} + ... + {}^6{C_6}.{}^6{C_0}\]
= 12C6 = 924 Answer the question to unlock