Question

If the value of the sum $29\binom{30}{0}+28\binom{30}{1}+27\binom{30}{2}+.....+1\binom{30}{28}+0\binom{30}{29}-\binom{30}{30}$ where $\begin{pmatrix} n \\ r \end{pmatrix}={ }^{ n } C _{ r }$, is equal to $k 2^{32}$, then the value of $k$ is equal to

• A. 7
• B. 14
• C. $\frac{5}{2}$
• D. $\frac{7}{2}$

Answer 1

Answer: (D)

$S= 29\binom{30}{0}+28\binom{30}{1}+27\binom{30}{2}+.....+1\binom{30}{28}+0\binom{30}{29}-\binom{30}{30} $
Also $S= -\binom{30}{0}+0\binom{30}{1}+1\binom{30}{2}+.......+27\binom{30}{28}+28\binom{30}{29}+29\binom{30}{30}$
$\Rightarrow 2 S =28\left({ }^{30} C _0+{ }^{30} C _1+{ }^{30} C _2+\ldots \ldots \ldots+{ }^{30} C _{30}\right)=28\left(2^{30}\right)$
Hence, $S =14 \times 2^{30}=\frac{7}{2} \times 2^{32} . \Rightarrow k =\frac{7}{2}$