The value of

Question:

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

(a) 27 − 1

(b) 28 − 2

(c) 28 − 1

(d) 28

Solution:

(b) $2^{8}-2$

$\left({ }^{7} C_{0}+{ }^{7} C_{1}\right)+\left({ }^{7} C_{1}+{ }^{7} C_{2}\right)+\left({ }^{7} C_{2}+{ }^{7} C_{3}\right)+\left({ }^{7} C_{3}+{ }^{7} C_{4}\right)+\left({ }^{7} C_{4}+{ }^{7} C_{5}\right)+\left({ }^{7} C_{5}+{ }^{7} C_{6}\right)+\left({ }^{7} C_{6}+{ }^{7} C_{7}\right)$

$=1+2 \times{ }^{7} C_{1}+2 \times{ }^{7} C_{2}+2 \times{ }^{7} C_{3}+2 \times{ }^{7} C_{4}+2 \times{ }^{7} C_{5}+2 \times{ }^{7} C_{6}+1$

$=1+2 \times{ }^{7} C_{1}+2 \times{ }^{7} C_{2}+2 \times{ }^{7} C_{3}+2 \times{ }^{7} C_{3}+2 \times{ }^{7} C_{2}+2 \times{ }^{7} C_{6}+1$

$=2+2^{2}\left({ }^{7} C_{1}+{ }^{7} C_{2}+{ }^{7} C_{3}\right)$

$=2+2^{2}\left(7+\frac{7}{2} \times 6+\frac{7}{3} \times \frac{6}{2} \times 5\right)$

$=2+252$

$=254$

$=2^{8}-2$

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