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Question Number 179948 by mr W last updated on 04/Nov/22

20 s t u d e n t s s h o u l d s t a n d i n 5

d i f f e r e n t r o w s . e a c h r o w s h o u l d h a v e

a t l e a s t 2 s t u d e n t s . i n h o w m a n y w a y s

c a n y o u a r r a n g e t h e m ?

(a n u n s o l v e d o l d q u e s t i o n)

Commented by mr W last updated on 04/Nov/22

f o r t e s t p u r p o s e w e c a n t a k e 10

s t u d e n t s i n 3 r o w s .

Commented by JDamian last updated on 04/Nov/22

are the students distinguishable?

Commented by mr W last updated on 04/Nov/22

y e s, w e {d o n}^{'} t h a v e i d e n t i c a l s t u d e n t s .

Answered by Frix last updated on 04/Nov/22

5 rows

2 2 2 2 12 \times \frac{5!}{4!} 2 2 2 3 11 \times \frac{5!}{3!} 2 2 2 4 10 \times \frac{5!}{3!}

2 2 2 5 9 \times \frac{5!}{3!} 2 2 2 6 8 \times \frac{5!}{3!} 2 2 2 7 7 \times \frac{5!}{3! 2!}

2 2 3 3 10 \times \frac{5!}{2!^{2}} 2 2 3 4 9 \times \frac{5!}{2!} 2 2 3 5 8 \times \frac{5!}{2!}

2 2 3 6 7 \times \frac{5!}{2!} 2 2 4 4 8 \times \frac{5!}{2!^{2}} 2 2 4 5 7 \times \frac{5!}{2!}

2 2 4 6 6 \times \frac{5!}{2!^{2}} 2 2 5 5 6 \times \frac{5!}{2!^{2}} 2 3 3 3 9 \times \frac{5!}{3!}

2 3 3 4 8 \times \frac{5!}{2!} 2 3 3 5 7 \times \frac{5!}{2!} 2 3 3 6 6 \times \frac{5!}{2!^{2}}

2 3 4 4 7 \times \frac{5!}{2!} 2 3 4 5 6 \times 5! 2 3 5 5 5 \times \frac{5!}{3!}

2 4 4 4 6 \times \frac{5!}{3!} 2 4 4 5 5 \times \frac{5!}{2!^{2}} 3 3 3 3 8 \times \frac{5!}{4!}

3 3 3 4 7 \times \frac{5!}{3!} 3 3 3 5 6 \times \frac{5!}{3!} 3 3 4 4 6 \times \frac{5!}{2!^{2}}

3 3 4 5 5 \times \frac{5!}{2!^{2}} 3 4 4 4 5 \times \frac{5!}{3!} 4 4 4 4 4 \times \frac{5!}{5!}

1001 possibilities

20 students = 20! possibilities

total :

1001 \times 20! = 2 435 334 910 184 816 640 000

Commented by mr W last updated on 05/Nov/22

y e s, {i t}^{'} s c o r r e c t, t h a n k s s i r!

Commented by mr W last updated on 05/Nov/22

t h e n u m b e r i s s o t e r r i b l y h u g e!

t h e q u e s t i o n i s t h e s a m e a s i n h o w

m a n y w a y s w e c a n a r r a n g e 20 b o o k s

i n a b o o k r a c k w i t h 5 s h e l v e s .

s a y w e c a n w o r k v e r y q u i c k l y a n d

n e e d o n l y o n e s e c o n d f o r e a c h

a r r a n g e m e n t, a n d w e b e g a n t o w o r k

a s t h e u n i v e r s e b e g a n t o e x i s t, i . e .

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

t h i n k ? h a v e w e d o n e a l l t h e p o s s i b l e

a r r a n g e m e n t s ?

n o! n o b y f a r! i n f a c t w e {h a v e n}^{'} t e v e n

d o n e 0 . 02 % o f t h e w o r k y e t!

n o t t o t h i n k, w h a t w h e n w e h a v e

100 b o o k s!

Commented by mr W last updated on 05/Nov/22

Answered by mr W last updated on 05/Nov/22

s a y t h e r o w s a r e l a b e l e d a s A, B, C, D, E .

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

a r e a, b, c, d, e r e s p e c t i v e l y .

a + b + c + d + e = 20

2 ⩽ a, b, c, d, e

t h e n u m b e r o f w a y s t o d i s t r i b u t e 20

p e o p l e i n t o 5 g r o u p s w i t h a, b, c, d, e

p e o p l e r e s p e c t i v e l y i s :

\frac{20!}{a! b! c! d! e!}

t o a r r a n g e t h e p e o p l e i n t h e i r r o w s

t h e r e a r e a! b! c! d! e! w a y s .

\frac{20!}{a! b! c! d! e!} \times a! b! c! d! e! = 20!

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

20! {(x^{2} + x^{3} + \dots)}^{5} . t h e a n s w e r o f t h e

q u e s t i o n i s t h e c o e f . o f x^{20} i n t h e

e x p a n s i o n o f

20! {(x^{2} + x^{3} + x^{4} + \dots)}^{5}

= \frac{20! x^{10}}{{(1 - x)}^{5}}

= 20! x^{10} \underset{k = 0}{\sum^{\infty}} C_{4}^{k + 4} x^{k}

10 + k = 20 \Rightarrow k = 10

s o t h e c o e f . o f x^{20} i s 20! C_{4}^{14} .

g e n e r a l l y f o r n p e o p l e i n r r o w s t h e

a n s w e r i s n! C_{r - 1}^{n - r - 1} .

i n c a s e o f 10 p e o p l e i n 3 r o w s, t h e

a n s w e r i s

10! C_{2}^{6} = 54 432 000

t h i s c a n b e m a n u a l l y c h e c k e d :

6 + 2 + 2 \Rightarrow \frac{10!}{6! {(2!)}^{2} 2!} \times 3! \times 6! \times 2! \times 2! = \frac{10! 3!}{2!}

5 + 3 + 2 \Rightarrow \frac{10!}{5! 3! 2!} \times 3! \times 5! \times 3! \times 2! = 10! \times 3!

4 + 4 + 2 \Rightarrow \frac{10!}{{(4!)}^{2} 2! 2!} \times 3! \times 4! \times 4! \times 2! = \frac{10! 3!}{2!}

4 + 3 + 3 \Rightarrow \frac{10!}{4! {(3!)}^{2} 2!} \times 3! \times 4! \times 3! \times 3! = \frac{10! 3!}{2!}

\Rightarrow 3 \times \frac{10! 3!}{2!} + 10! 3! = 54 432 000 ✓

Commented by mr W last updated on 05/Nov/22

a l t e r n a t i v e w a y i n s t e a d o f u s i n g

g e n e r a t i n g f u n c t i o n :

a + b + c + d + e = 20

2 ⩽ a, b, c, d, e

l e t a = a^{'} + 1, b = b^{'} + 1, \dots

a^{'} + b^{'} + c^{'} + d^{'} + e^{'} = 15

1 ⩽ a^{'}, b^{'}, c^{'}, d^{'}, e^{'}

★ ∣ ★ ∣ ★ ∣ ★ ∣ ★ ∣ ★ ∣ ★ \dots ★ ∣ ★ ∣ ★

(15 s t a r s a n d 14 b a r s)

t h e n u m b e r o f w a y s t o s e l e c t 4 o f t h e

14 b a r s i s C_{4}^{14} = 1001 . s o t h e a n s w e r i s

20! C_{4}^{14} = 1001 \times 20!

Commented by Frix last updated on 05/Nov/22

thank you for explaining

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