the value of (((50)),(0) )^2 + (((50)),(1) )^2 + (((50)),(2) )^2 +...+ (((50)),((49)) )^2 + (((50)),((50)) )^2


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Question Number 151073 by mathdanisur last updated on 18/Aug/21

$the value of$ ${(\begin{matrix} 50 \\ 0 \end{matrix})}^{2} + {(\begin{matrix} 50 \\ 1 \end{matrix})}^{2} + {(\begin{matrix} 50 \\ 2 \end{matrix})}^{2} + . . . + {(\begin{matrix} 50 \\ 49 \end{matrix})}^{2} + {(\begin{matrix} 50 \\ 50 \end{matrix})}^{2}$
	Answered by ArielVyny last updated on 18/Aug/21

	$S_{n} = \underset{k = 0}{\sum^{50}} {(C_{50}^{k})}^{2}$ $l n (S_{n}) = 2 \underset{k = 0}{\sum^{50}} l n (\frac{50!}{k! (50 - k)!}) = 2 l n (50!) (50 + 1) - 2 \underset{k = 0}{\sum^{50}} l n [k! (50 - k)!]$ $l n (S_{n}) = 102 l n (50!) - 2 \underset{k = 0}{\sum^{50}} l n (k!) - 2 \underset{k = 0}{\sum^{50}} l n (50 - k)!$ $l n (S_{n}) = 102 l n (50!) \to S_{n} = e^{102 l n (50!)} = e^{l n {(50!)}^{102}}$ $S_{n} = {(50!)}^{102}$
	Commented by mathdanisur last updated on 18/Aug/21

	$Thank you S er, answer : (\begin{matrix} 100 \\ 50 \end{matrix})$
	Commented by mr W last updated on 18/Aug/21

	$w r o n g!$ $S = a + b$ $\ln S = \ln (a + b) \neq \ln a + \ln b$
	Answered by mr W last updated on 18/Aug/21

	$i n {(1 + x)}^{n} {(1 + x)}^{n}$ $t h e c o e f . o f x^{n} i s \underset{k = 0}{\sum^{n}} C_{k}^{n} C_{n - k}^{n} = \underset{k = 0}{\sum^{n}} {(C_{k}^{n})}^{2}$ $b u t {(1 + x)}^{n} {(1 + x)}^{n} = {(1 + x)}^{2 n} a n d t h e$ $t h e c o e f . o f x^{n} i s C_{n}^{2 n} . t h e r e f o r e$ $\underset{k = 0}{\sum^{n}} {(C_{k}^{n})}^{2} = C_{n}^{2 n}$ $\Rightarrow {(C_{0}^{50})}^{2} + {(C_{1}^{50})}^{2} + . . . + {(C_{50}^{50})}^{2} = C_{50}^{100}$
	Commented by mathdanisur last updated on 18/Aug/21

	$Thank you Ser$
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