13456622577532674 how many 5-digit numbers can be made from these numbers? help please


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Question Number 209794 by Ismoiljon_008 last updated on 21/Jul/24

$13456622577532674 h o w m a n y 5 - d i g i t$ $n u m b e r s c a n b e m a d e f r o m t h e s e n u m b e r s ?$ $h e l p p l e a s e$
	Answered by MM42 last updated on 21/Jul/24

	$1 \to 1 / 2 \to 3 / 3 \to 2 / 4 \to 2 / 5 \to 3 / 6 \to 3 / 7 \to 3$ $a b c d e \Rightarrow 7 \times 6 \times 5 \times 4 \times 3 = 2520$ $a a b c d \Rightarrow (\begin{matrix} 6 \\ 1 \end{matrix}) \times (\begin{matrix} 6 \\ 3 \end{matrix}) \times \frac{5!}{2!} = 7200$ $a a b b c \Rightarrow (\begin{matrix} 6 \\ 2 \end{matrix}) \times (\begin{matrix} 5 \\ 1 \end{matrix}) \times \frac{5!}{2! 2!} = 2250$ $a a a b c \Rightarrow (\begin{matrix} 4 \\ 1 \end{matrix}) \times (\begin{matrix} 6 \\ 2 \end{matrix}) \times \frac{5!}{3!} = 1200$ $a a a b b \Rightarrow (\begin{matrix} 4 \\ 1 \end{matrix}) \times (\begin{matrix} 5 \\ 1 \end{matrix}) \times \frac{5!}{3! 2!} = 200$ $a n s = 13370 ✓$
	Commented by Ismoiljon_008 last updated on 22/Jul/24

	$t h a n k y o u v e r y m u c h$
	Commented by mr W last updated on 22/Jul/24

	$m e t h o d i s r i g h t, b u t a n s w e r i s w r o n g .$ $a b c d e \Rightarrow 7 \times 6 \times 5 \times 4 \times 3 = 2520 ✓$ $a a b c d \Rightarrow (\begin{matrix} 5 \\ 1 \end{matrix}) \times (\begin{matrix} 6 \\ 3 \end{matrix}) \times \frac{5!}{2!} = 6000$ $a a b b c \Rightarrow (\begin{matrix} 5 \\ 2 \end{matrix}) \times (\begin{matrix} 5 \\ 1 \end{matrix}) \times \frac{5!}{2! 2!} = 1500$ $a a a b c \Rightarrow (\begin{matrix} 4 \\ 1 \end{matrix}) \times (\begin{matrix} 6 \\ 2 \end{matrix}) \times \frac{5!}{3!} = 1200 ✓$ $a a a b b \Rightarrow (\begin{matrix} 4 \\ 1 \end{matrix}) \times (\begin{matrix} 4 \\ 1 \end{matrix}) \times \frac{5!}{3! 2!} = 160$ $t o t a l l y : 11380$
	Answered by mr W last updated on 22/Jul/24

	$a n o t h e r p a t h u s i n g g e n e r a t i n g f u n c t i o n :$ $1, 3$ $44$ $222, 555, 666, 777$ $t h e a n s w e r i s t h e c o e f . o f t e r m x^{5} o f$ $t h e e x p a n s i o n$ $5! {(1 + x)}^{2} (1 + x + \frac{x^{2}}{2!}) {(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!})}^{4}$ $w h i c h i s 11380 .$
	Commented by mr W last updated on 22/Jul/24

	Commented by mr W last updated on 22/Jul/24

	$i f t h e q u e s t i o n i s ‘ ‘ h o w m a n y 7$ $d i g i t n u m b e r s c a n b e {m a d e}^{″}, t h e n$ $t h e a n s w e r i s t h e c o e f . o f t e r m x^{7} o f$ $7! {(1 + x)}^{2} (1 + x + \frac{x^{2}}{2!}) {(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!})}^{4}$ $w h i c h i s 358680 .$
	Commented by mr W last updated on 22/Jul/24

	Commented by mr W last updated on 22/Jul/24

	$o r w e j u s t c o n s i d e r t h e g e n e r a t i n g$ $f u n c t i o n$ ${(1 + x)}^{2} (1 + x + \frac{x^{2}}{2!}) {(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!})}^{4}$ $t h e n t h e n u m b e r o f k - d i g i t n u m b e r s$ $i s k! \times c o e f f i c i e n t o f t e r m x^{k} .$
	Commented by mr W last updated on 22/Jul/24

	Commented by mr W last updated on 22/Jul/24

	$e x a m p l e s :$ $4 d i g i t n u m b e r s : 4! \times 79 = 1896$ $5 d i g i t n u m b e r s : 5! \times \frac{569}{6} = 11380$ $6 d i g i t n u m b e r s : 6! \times \frac{1091}{12} = 65460$ $7 d i g i t n u m b e r s : 7! \times \frac{427}{6} = 358680$ $10 d i g i t n u m b e r s : 10! \times \frac{9577}{864} = 40223400$ $e t c .$
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