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

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150 ?
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0
13?
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This is similar to total number of onto functions from m elements to n elements.


$\sum_{k=0}^{n}(-1)^{k}$ $^{n}C_{k}(n-k)^{m}$

Given m=5 and n=3

$\therefore$  The total number of ways in which 5 balls of different color can be  distributed among 3 persons so that each person gets at least one ball is

= $3^{5}-$ $^{3}C_{1}(3-1)^{5}+$  $^{3}C_{2}(3-2)^{5}-$  $^{3}C_{3}(3-3)^{5}$

= 243 - 32*3 + 3 - 1

=243 - 93

=150

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