The value of C 15 0 2 - C 15 1 2 + C 15 2 2 - ? - C 15 15 2 is

Question

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

Smart Achievers · Accepted Answer

(1)

$Let f (n) = {{}^{n}C}_{0}^{2} - {{}^{n}C}_{1}^{2} + {{}^{n}C}_{2}^{2} - {{}^{n}C}_{3}^{2} + \dots + {(- 1)}^{n} {{}^{n}C}_{n}^{2}$

$∵$ $f (n) =$ ${\begin{matrix} 0, & n = odd \\ {(- 1)}^{n / 2}, {}^{n}C_{n / 2}, & n = even \end{matrix}$

Here, $n = 15$ (odd)

$∴$ ${{}^{15}C}_{0}^{2} - {{}^{15}C}_{1}^{2} + {{}^{15}C}_{2}^{2} - \dots - {{}^{15}C}_{15}^{2} = 0$

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