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Question Number 156511 by KONE last updated on 12/Oct/21

	Answered by KONE last updated on 12/Oct/21

	$s v p a i d e d a n s l a q u e s t i o n 2$
	Answered by mindispower last updated on 12/Oct/21

	$\frac{n!}{k!} = \underset{j = k + 1}{\prod^{n}} j$ $\underset{k = 0}{\prod^{n - 1}} \underset{k + 1}{\prod^{n}} j = \underset{k = 0}{\prod^{n - 1}} (k + 1 . . . . . . n)$ $= (1 . . . . n) . (2 . . . . . n) (3 . . . . . . n) . (4 . . . . n) . . . . . n$ $= n^{n} . {(n - 1)}^{n - 1} . . . . . . . . 2^{2} . 1^{1} = \prod_{k ⩽ n} k^{k}$
	Commented by puissant last updated on 12/Oct/21

	$G o o d s i r M i n d i s p o w e r . . .$ $M r K o n e c^{'} e s t b i e n d e d i r e m e r c i p o u r$ $m o n t r e r a c e l u i q u i t^{'} a a i d \overset{´}{e} q u e t u a s$ $v u l a r e p o n s e e t q u e t u a p p r e c i e s . . .$ $c a f e r r a e n s o r t e q u e p r o c h a i n e m e n t$ $o n t^{'} a i d e f a c i l e m e n t . . .$ $b o n n e c o n t i n u a t i o n . . .$
	Commented by KONE last updated on 13/Oct/21

	$s a l u t a v o u s j e m^{'} e x c u s e m a c o n n e x i o n n^{'} e t a i s p a s t r o p e n p l a c e$ $m e r c i b i e n a v o u s p o u r v o t r e a s s i s t a n c e$
	Commented by KONE last updated on 13/Oct/21

	$s v p j e n e c o m p r e n d s a s t r o p l a p r e m i e r e l i g n e$
	Commented by puissant last updated on 14/Oct/21

	$R e m a r q u e q u e l e p r o d u i t c o m m e n c e a$ $k + 1 e n f a i t c a v a d e v e n i r$ $\underset{j = k + 1}{\prod^{n}} = (k + 1) (k + 2) . . . . n o r p o u r e c r i r e f a c t o r i e l$ $n, i l f a u t c o m m e n c e r a 1 c e t a d i r e$ $1.2.3 . . . . n m a i n t e n a n t c o m m e o n a p a s c a,$ $o n a j o u t e 1.2.3 . . . k = k! e n h a u t e t e n$ $b a s c e t a d i r e n (n - 1) . . . . (k + 1) \frac{k (k - 1) . . . 2.1}{k (k - 1) . . . 2.1}$ $c a d e v i e n t a l o r s :$ $\frac{n (n - 1) . . . (k + 1) k (k - 1) . . . . 2.1}{k (k - 1) (k - 2) . . . . 2.1} = \frac{n!}{k!}$ $b r e f c^{'} e s t j u s t e u n p e t i t j e u s u r l e p r o d u i t$ ${q u}^{'} i l f a u t r e m a r q u e r . . .$ $C o r d i a l e m e n t . .$
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