Question

Tom has $15$ -ping-pong balls each uniquely numbered from 1 to 15. He also has a red box, a blue box, and a green box.
a) How many ways Tom place the $15$ distinct balls in to the three boxes so that no box is empty?
b) Suppose now that Toms has placed $5$ ping-pong balls in each box. How many ways can he choose 5 balls from the three boxes so that he choose at least on from each box?

Answer 1

Solution-

All balls are distinct.

i) Let Tom places x balls in red, y balls in blue, z

balls in green box

then we have to find positive integral solution of

$x+y+z=15$

Or $x+y+z=12\to$ Non negative integral solutions

Total ways ${=}^{3+12-1}{C}_{12-1}{=}^{14}{C}_{11}$

to distribute

Total ways with arranging $15!{\times }^{14}{C}_{11}$

ii) Let be chooses x balls from red box, y balls from

blue 10x, z balls form green 10x

then

No negative integral solution of $x+y+z=2$

Total ways ${=}^5{C}_1{\times }^5{C}_1{\times }^5{C}_1{\times }^{3+2-1}{C}_{2-1}$

$={125}^4{C}_1=500.$