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  4. The value of ( 7 C0+ 7 C1)+( 7 C1+ 7 C2)+⋅s+( 7 C6+ 7 C7) is

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Q. The value of $\left({ }^{7} C_{0}+{ }^{7} C_{1}\right)+\left({ }^{7} C_{1}+{ }^{7} C_{2}\right)+\cdots+\left({ }^{7} C_{6}+{ }^{7} C_{7}\right)$ is

Binomial Theorem

Solution:

$\left({ }^{7} C_{0}+{ }^{7} C_{1}\right)+\left({ }^{7} C_{1}+{ }^{7} C_{2}\right)+\cdots+\left({ }^{7} C_{6}+{ }^{7} C_{7}\right)$
$={ }^{8} C_{1}+{ }^{8} C_{2}+\cdots+{ }^{8} C_{7}={ }^{8} C_{0}+{ }^{8} C_{1}+{ }^{8} C_{2}+\cdots+{ }^{8} C_{7}+{ }^{8} C_{8}$
$-\left({ }^{8} C_{0}+{ }^{8} C_{8}\right)=2^{8}-1(1+1)=2^{8}-2$