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

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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