Evaluate $C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - . . . . . . + (- 1)^{n} C_{n}^{2}$ for n = 10 and n = 11

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Solution

For n = 10 the value ,
$= (- 1)^{10 / 2}^{10} C_{5} = (- 1)^{5} \frac{(10)!}{5! 5!}$
$= (- 1)^{5} \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = - 252$
And for n = 11 , the value = 0 as 11/2 is not an integer

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