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  • Find the value of <img alt="\sum_{r=1}^{5}\left ( \binom{5}{r} \right )^{2}" src="http://entrancecorner.oncodecogs.com/gif.latex?%5Csum_%7Br%3D1%7D%5E%7B5%7D%5Cleft%20%28%20%5Cbinom%7B5%7D%7Br

Find the value of \sum_{r=1}^{5}\left ( \binom{5}{r} \right )^{2}

Option: 1

\binom{8}{4}


Option: 2

\binom{10}{5}


Option: 3

\binom{10}{4}


Option: 4

None of these


Answers (1)

best_answer

As we learnt

Series Involving Product of two Binomial Coefficients

If the difference of lower suffixes of binomial coefficients in each term is same (=k)

^nC_0. ^mC_k +^nC_1. ^mC_{k+1} +^nC_2. ^mC_{k+2} +.......... = ^{m+n} C _{n-k}

 

Now,

 \sum_{r=1}^{5}\binom{5}{r}^{2}= \binom{5}{1}^2+\binom{5}{2}^2+...+\binom{5}{5}^2

= \left (\binom{5}{0}^2+\binom{5}{1}^2+...+\binom{5}{5}^2 \right )-1

=\binom{10}{5} -1

 

Posted by

avinash.dongre

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