define Σ_(x=0) ^k ((n!)/(x!(n−x)!)) without Σ symbol


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Question Number 93764 by DuDono last updated on 14/May/20

$define$ $\underset{x = 0}{\sum^{k}} \frac{n!}{x! (n - x)!}$ $without Σ symbol$
Commented by mr W last updated on 14/May/20

$d o y o u m e a n \underset{x = 0}{\sum^{n}} \frac{n!}{x! (n - x)!} ?$ $\underset{x = 0}{\sum^{n}} \frac{n!}{x! (n - x)!} = \underset{x = 0}{\sum^{n}} C_{x}^{n} = {(1 + 1)}^{n} = 2^{n}$
Commented by DuDono last updated on 14/May/20

$no, x must appear somewhere . You$ $can imagine it as a function if you want$
Commented by mr W last updated on 14/May/20

$T H E N Y O U R Q U E S T I O N I S$ $W R O N G!$ $i n \underset{x = 0}{\sum^{k}} a_{x} t h e s y m b o l x i s j u s t a c o u n t e r,$ $y o u c a n r e p l a c e i t w i t h a n y s y m b o l .$ $\underset{x = 0}{\sum^{k}} a_{x} = \underset{y = 0}{\sum^{k}} a_{y} = \underset{ψ = 0}{\sum^{k}} a_{ψ} .$ $\underset{x = 0}{\sum^{k}} \frac{n!}{x! (n - x)!} c a n b e a f u n c t i o n o f n$ $o r a f u n c t i o n o f k, b u t n e v e r a$ $f u n c t i o n o f x . t h i s i s s u c h b a s i c i n$ $m a t h e m a t i c s a s 1 + 1 = 2 .$
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