find (C_0 ^(100) )^2 +(C_2 ^(100) )^2 +(C_4 ^(100) )^2 +(C_6 ^(100) )^2 +...+(C


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Question Number 157628 by mr W last updated on 25/Oct/21

$f i n d$ ${(C_{0}^{100})}^{2} + {(C_{2}^{100})}^{2} + {(C_{4}^{100})}^{2} + {(C_{6}^{100})}^{2} + . . . + {(C_{100}^{100})}^{2} = ?$
	Answered by mindispower last updated on 25/Oct/21

	$\underset{k = 0}{\sum^{100}} {(C_{2 k}^{100})}^{2}$ ${(1 + i x)}^{100} {(1 - i x)}^{100} = \underset{k = 0}{\sum^{100}} C_{100}^{k} {(i x)}^{k} \underset{n = 0}{\sum^{100}} C_{n}^{100} {(- i x)}^{n}$ $= {(1 + x^{2})}^{100} = \underset{k = 0}{\sum^{100}} C_{100}^{k} {(i x)}^{k} . \underset{n = 0}{\sum^{100}} C_{n}^{100} {(- i x)}^{n}$ $n + k = 100$ $\Rightarrow C_{100}^{50} x^{100} = \underset{k = 0}{\sum^{100}} C_{100}^{k} {(i x)}^{k} . C_{100 - k}^{100} {(- i)}^{100 - k} x^{100 - k}$ $\Rightarrow \underset{k = 0}{\sum^{100}} C_{k}^{100} . C_{100 - k}^{100} {(- 1)}^{k} x^{100}$ $\Rightarrow \underset{k = 0}{\sum^{100}} {(- 1)}^{k} {(C_{k}^{100})}^{2} = C_{50}^{100}$ ${(1 + x)}^{100} {(1 + x)}^{100} = \underset{k = 0}{\sum^{100}} C_{k}^{100} x^{k} \underset{n = 0}{\sum^{100}} C_{n}^{100} x^{n}$ $C_{100}^{200} x^{100} = \underset{k = 0}{\sum^{100}} {(C_{k}^{100})}^{2}$ $Σ {(C_{2 k}^{100})}^{2} = \frac{1}{2} \underset{k = 0}{\sum^{100}} ({(C_{k}^{100})}^{2} + {(- 1)}^{k} {(C_{k}^{100})}^{2})$ $\underset{k = 0}{\sum^{50}} (C_{2 k}^{100}) = \frac{1}{2} (C_{100}^{200} + C_{50}^{100})$
	Commented by mr W last updated on 26/Oct/21

	$g r e a t! t h a n k s s i r!$
	Commented by mindispower last updated on 26/Oct/21

	$w i t h e p l e a s u r s i r$ $H a v e a n i c e d a y$
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