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Question Number 212686 by mr W last updated on 21/Oct/24

i n h o w m a n y w a y s c a n a t e a c h e r

d i v i d e h i s 10 s t u d e n s i n t o 4 g r o u p s

s u c h t h a t e a c h g r o u p h a s a t l e a s t 2

s t u d e n t s ?

Commented by Spillover last updated on 21/Oct/24

2268000 ?

Commented by mr W last updated on 21/Oct/24

h o w ?

Commented by Spillover last updated on 21/Oct/24

i s i t c o r r e c t ?

Commented by Spillover last updated on 21/Oct/24

\frac{10!}{{(2!)}^{4}} = 2268000

Answered by mehdee7396 last updated on 21/Oct/24

2 / 2 / 2 / 4 o r 2 / 2 / 3 / 3

\Rightarrow (\begin{matrix} 10 \\ 2, 2, 2, 4 \end{matrix}) + (\begin{matrix} 10 \\ 2, 2, 3, 3 \end{matrix}) = 44100

Commented by mr W last updated on 21/Oct/24

y o u r a n s w e r w e r e c o r r e c t i f t h e

q u e s t i o n i s t o d i v i d e t h e s t u d e n t s

i n t o 4 g r o u p s f o r m a t h m a t i c s,

p h y s i c s, c h e m i s t r y a n d b i o l o g y

r e s p e c t i v e l y, i . e . d i s t i n c t o b j e c t s

i n t o d i s t i n c t b i n s . b u t i n t h i s q u e s t i o n

w e j u s t d i v i d e 10 s t u d e n t s i n t o 4

(i d e n t i c a l) g r o u p s .

Commented by mehdee7396 last updated on 21/Oct/24

y o u a r e r i g h t \underset{―}{\underset{⏟}{⋚}}

Answered by mr W last updated on 21/Oct/24

2 / 2 / 2 / 4 \Rightarrow \frac{10!}{{(2!)}^{3} 3! 4!} = 3150

2 / 2 / 3 / 3 \Rightarrow \frac{10!}{{(2!)}^{2} 2! {(3!)}^{2} 2!} = 6300

Σ : 3150 + 6300 = 9450

Answered by Spillover last updated on 21/Oct/24

S t u d e n t = 10 (n) s a y

g r o u p = 4 (k) s a y

s e l e c t e d s t u d e n t i n e a c h g r o u p = (2 \times 4) = 8

10 - 8 = 2

N u m b e r o f a r r a n g e m e n t = (\begin{matrix} n + k - 1 \\ k - 1 \end{matrix})

(\begin{matrix} n + k - 1 \\ k - 1 \end{matrix}) = (\begin{matrix} 2 + 4 - 1 \\ 4 - 1 \end{matrix}) = (\begin{matrix} 5 \\ 3 \end{matrix}) = 10

T o t a l w a y = \frac{10!}{{(2!)}^{4}} \times 10 = 2268000

Commented by mr W last updated on 21/Oct/24

t h i s i s w r o n g s i r .

t o d i v i d e 10 d i s t i n c t o b j e c t s i n t o 4

i d e n t i c a l b i n s s u c h t h a t n o b i n i s

e m p t y, t h e r e a r e {_{4}^{10}} = S (10, 4) = 34105

w a y s . s i n c e i n o u r q u e s t i o n e a c h b i n

s h o u l d h a v e a t l e a s t 2 o b j e c t s, s o t h e

a n s w e r s h o u l d b e l e s s t h a n 34105 .

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

{_{4}^{10}} o r S (10, 4) i s t h e s t i r l i n g

n u m b e r o f t h e s e c o n d k i n d .

Answered by golsendro last updated on 22/Oct/24

(1) \frac{(\begin{matrix} 10 \\ 2 \end{matrix}) (\begin{matrix} 8 \\ 2 \end{matrix}) (\begin{matrix} 6 \\ 2 \end{matrix}) (\begin{matrix} 4 \\ 4 \end{matrix})}{3!}

(2) \frac{(\begin{matrix} 10 \\ 2 \end{matrix}) (\begin{matrix} 8 \\ 2 \end{matrix}) (\begin{matrix} 6 \\ 3 \end{matrix}) (\begin{matrix} 3 \\ 3 \end{matrix})}{{(2!)}^{2}}

Commented by golsendro last updated on 22/Oct/24

yes