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 one ball is [2012]

Question

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 one ball is [2012]

Smart Achievers · Accepted Answer

(2)

$∵$ Each person gets at least one ball.

$∴$ 3 persons can have 5 balls as follows.

Person	No. of balls	No. of balls
I	1	1
II	1	2
III	3	2

The number of ways to distribute balls 1,1,3 in first to three persons $= {}^{5}C_{1} \times {}^{4}C_{1} \times {}^{3}C_{3}$

Also 3 persons having 1,1 and 3 balls can be arranged in $\frac{3!}{2!}$ ways.

$∴$ Total number of ways to distribute 1,1,3 balls to the three persons $= {}^{5}C_{1} \times {}^{4}C_{1} \times {}^{3}C_{3} \times \frac{3!}{2!} = 60$

Similarly, total number of ways to distribute 1, 2, 2 balls to three persons $= {}^{5}C_{1} \times {}^{4}C_{2} \times {}^{2}C_{2} \times \frac{3!}{2!} = 90$

$∴$ $The required number of ways = 60 + 90 = 150$

Solution

Q.
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 one ball is [2012]

1 75

2 150

3 210

4 243

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