The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least on ball is

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Solution

$\begin{matrix} T o t a l n o . o f b a l l s = 5 N o . o f p e r s o n = 3 E a c h p e r s o n s h o u l d g e t a t l e a s t o n e b a l l . ∴ T h e s e l e c t i o n o f b a l l s c a n b e d o n e b y (1, 1, 3) o r (2, 2, 1) i . e ., (^{5} C_{1} \times^{4} C_{1} \times^{3} C_{1}) o r (^{5} C_{2} \times^{3} C_{2} \times^{1} C_{1}) = (5 \times 4 \times 1) o r (10 \times 3 \times 1) = 20 o r 30 w a y s A f t e r s e l e c t i o n i t s h o u l d b e d i s t r i b u t e d b t w e e n t h r e e d i f f e r e n t p e r s o n s . (1, 1, 3) s e l e c t i o n i s d i s t r i b u t e d i n \frac{3!}{2!} = 3 w a y s a n d, (2, 2, 1) s e l e c t i o n i s d i s t r i b u t e d i n \frac{3!}{2!} = 3 w a y s ∴ T h e r e q u i r e d n o . o f w a y s = (20 \times 3) + (30 \times 3) = 60 + 90 = 150. \end{matrix}$

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