Suppose-that-7-blue-balls-8-red-balls-and-9-green-balls-should-be-put-into-three-boxes-labeled-1-2-and-3-so-that-any-box-contains-at-least-one-balls-of-each-colour-How-many-ways-can-this-arrangeme

Question Number 119747 by benjo_mathlover last updated on 26/Oct/20

S u p p o s e t h a t 7 b l u e b a l l s, 8 r e d b a l l s a n d 9 g r e e n

b a l l s s h o u l d b e p u t i n t o t h r e e b o x e s l a b e l e d

1, 2 a n d 3, s o t h a t a n y b o x c o n t a i n s a t l e a s t

o n e b a l l s o f e a c h c o l o u r . H o w m a n y w a y s

c a n t h i s a r r a n g e m e n t b e d o n e ?

Answered by mr W last updated on 26/Oct/20

t o d i v i d e 7 b l u e b a l l s i n t o 3 g r o u p s,

u s i n g s t a r s & b a r s m e t h o d,

∣ ∣ ∣ ∣ ∣ ∣

t h e r e a r e C_{2}^{6} w a y s .

s i m i l a r l y t o d i v i d e 8 r e d b a l l s i n t o

3 g r o u p s t h e r e a r e C_{2}^{7} w a y s, a n d t o

d i v i d e 9 g r e e n b a l l s i n t o 3 g r o u p s

t h e r e a r e C_{2}^{8} w a y s, s o t o t a l l y w e h a v e

C_{2}^{6} \times C_{2}^{7} \times C_{2}^{8} = 8820 w a y s .

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