sum_{r=0}^{6}left({ }^{6} C _{r} cdot{ }^{6} | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

$\sum_{ r =0}^{6}{ }^{6} C _{ r } \cdot{ }^{6} C _{6- r }$

$={ }^{6} C _{0} \cdot{ }^{6} C _{6}+{ }^{6} C _{1} \cdot{ }^{6} C _{5}+\ldots \ldots+{

Vedclass · Accepted Answer

$924$

Vedclass · Answer

$\sum_{ r =0}^{6}{ }^{6} C _{ r } \cdot{ }^{6} C _{6- r }$

$={ }^{6} C _{0} \cdot{ }^{6} C _{6}+{ }^{6} C _{1} \cdot{ }^{6} C _{5}+\ldots \ldots+{ }^{6} C _{6} \cdot{ }^{6} C _{0}$

Now,

$(1+x)^{6}(1+x)^{6}$

$=\left(\begin{array}{l}\left.{ }^{6} C_{0}+{ }^{6} C_{1} x+{ }^{6} C_{2} x^{2}+\ldots . . .+{ }^{6} C_{6} x^{6}ight) \ \left({ }^{6} C_{0}+{ }^{6} C_{1} x+{ }^{6} C_{2} x^{2}+\ldots \ldots+{ }^{6} C_{6} x^{6}ight)\end{array}ight.$ 

Comparing coefficeint of $x^{6}$ both sides

$\begin{array}{l} { }^{6} C _{0} \cdot{ }^{6} C _{6}+{ }^{6} C _{1}+{ }^{6} C _{5}+\ldots \ldots .+{ }^{6} C _{6} \cdot{ }^{6} C _{0}={ }^{12} C _{6} \ =924 \end{array}$

$\sum_{r=0}^{6}\left({ }^{6} C _{r} \cdot{ }^{6} C _{6- r }\right)$ का मान बराबर है

Similar Questions

$\sum\limits_{k = 0}^{10} {^{20}{C_k} = } $

${(1 + x)^{15}}$ के प्रसार में अन्तिम आठ पदों के गुणांकों का योगफल है

यदि $\sum_{ r =0}^{25}\left\{{ }^{50} C _{ r } \cdot{ }^{50- r } C _{25- r }\right\}= K \left({ }^{50} C _{25}\right)$ हो, तो $K$ का मान होगा

यदि ${C_0},{C_1},{C_2},.......,{C_n}$ द्विपद गुणांक हो, तो $2.{C_1} + {2^3}.{C_3} + {2^5}.{C_5} + ....$ का $n$ पदों तक मान होगा