Question

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

Answer 1

let no. of balls each person gets be p1,p2,p3
then we have
p1 + p2 + p3 = 5
where p1 >=1
p2>=1
and p3>=1

okay,,before that,,
suppose we had this

a + b+c ....r = n
where each a,b,c >=0

no. of such arrangements = C(n+r -1 , r-1)
it'd be better if you directly rote down this formulla,,otherwise this thing can also be explained..needless for iit preparation though..

now..what we have is p1,p2,p3 >=1 and not 0

so we convert that
we can see
p1 - 1 >=0
p2- 1 >=0
p3 -1 =0

we had
p1+p2+p3 =5
or
(p1 -1) + (p2 -1) + (p3 -1) = 2

so no. of ways will be
C(2+3-1 , 3-1) =C(4,2) = 6