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Q. The value of $ \displaystyle \sum_{r=0}^{3}{ }^{8} C_{r}\left({ }^{5} C_{r+1}-{ }^{4} C_{r}\right)$ is____

Binomial Theorem

Solution:

$ \displaystyle\sum_{r=0}^{3}{ }^{8} C_{r}\left({ }^{5} C_{r+1}-{ }^{4} C_{r}\right)= \displaystyle\sum_{r=0}^{3}{ }^{8} C_{r}{ }^{4} C_{r+1}$
$={ }^{8} C_{0} \times{ }^{4} C_{1}+{ }^{8} C_{1} \times{ }^{4} C_{2}+{ }^{8} C_{2} \times{ }^{4} C_{3}+{ }^{8} C_{3} \times{ }^{4} C_{4}$
$=$ coefficient of $x^{3}$ in $(1+x)^{4}(1+x)^{8}$
$=$ coefficient of $x^{3}$ in $(1+x)^{12}$
$={ }^{12} C_{3}=220$