q) In how many ways can 10 different toys be distributed among 10 children so that exactly 2 of them get no toys.

my soln...

ways of selecting 2 children = 10C2

now all others should have atleast one...so 10C8*8! ways..

now the remaining 2 toys can be distributed in 8^2 ways..

so my ans is 10C2 * 10C2*8! * 8^2..

but this is not the right answer..plz point out the mistake in my answer and post the correct solution ..

I think the ans is 10_C2.10_C2.8!.9_C2.Your first two steps are correct,so,2 toys are left behind after giving one each to 8 members.This is similar to finding no. of solutions to

x₁+x₂+......+x₈=2 where x_i 0 for i=1,2,...,8 which gives (8+2-1)_C2=9_C2

Is my answer correct?

Sorry,I made a mistake.After the first two steps,two diff toys are left.select one toy and one person,the remaining toy is given by selecting another person or select both the toys and a person and give both to him only

thus,the ans is 10_C2.10_C2.8!.(2_C1.8_C1.7_C1+2_C2.8_C1)=120.10_C2.10_C2.8!

Am I correct?

see ans of joy is incorrect because he has considered the order in which the toys are given to the child as different cases

for eg: if a child has two toys after final distribution say toy 1 & toy 2 it should be one case but in joy's method it is two cases,

1st- toy1 from first 8 lot and toy2 from the other remaining 2 toys

2nd- toy 2 from the first 8 lot and toy 1 from the remaining two.

hence two eliminate this problem of order recurrence we shall take two cases

first case let the three toys be given to a single person

no.of ways 10C2 * 8C1 * 10C7 * 7!           

case 2 remaining two toys are given to two different persons(i.e 6 persons get one each and two get two toys each)

no.of ways 10C2 * 8C2 * 10C6 * 6! * 4C2

total ans  = case 1 + case2  

*edited*

well i think this is a simple q.

first select 2 childern 10C2 now @ joyfrancis

" now all others should have atleast one..." this statement is wrong.

there is no condition on the remaining so the ans. should be

10C2*8^8  joy is my ans matching i think it should.....