First slide

Combinations

Question

The sum $S$ = $\frac{1}{9!} + \frac{1}{3! 7!} + \frac{1}{5! 5!} + \frac{1}{7! 3!} + \frac{1}{9!}$ equals

Moderate

A

$\frac{2^{9}}{10!}$
B

$\frac{2^{10}}{8!}$
C

$\frac{2^{11}}{9!}$
D

$\frac{2^{10}}{7!}$

Solution

We can write $S$ as follows
$\begin{array}{l} S = \frac{1}{10!} [^{10} C_{1} +^{10} C_{3} +^{10} C_{5} +^{10} C_{7} +^{10} C_{9}] \\ = \frac{1}{10!} (2^{9}) \end{array}$

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