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Sum of Binomial Coefficients with Alternate Signs

The value of ...

Question

The value of $^{15} C_{0}^{2} -^{15} C_{1}^{2} +^{15} C_{2}^{2} - \dots -^{15} C_{15}^{2}$ is

A

15

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B

- 15

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C

0

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D

51

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Solution

The correct option is C $0$
Let $f (n) = C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - C_{3}^{2} + \dots + (- 1)^{n} C_{n}^{2}$
We know that
$f (n) = {\begin{matrix} 0, & if n is odd (- 1)^{n / 2} C_{n / 2}, & if n is even \end{matrix}$

Since, here $n$ is odd $(n = 15)$
$∴^{15} C_{0}^{2} -^{15} C_{1}^{2} +^{15} C_{2}^{2} - \dots -^{15} C_{15}^{2} = 0$

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Sum of Binomial Coefficients with Alternate Signs

Standard XII Mathematics

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