Question

1. Eight different gifts are to be distributed among three children, a) How many distinct results are possible? b) How many distinct results are possible if every child receives at least one gift? Hint. Note that "at least one child does not get anything" means "exactly one child does not get anything" or "exactly two do not get anything" 2. There are five flavors of ice cream: C, M, B, S and V a) How many variations of three scoop servings are there (order of the scoops does not matter different scoops could be of the same flavor)? b) In how many ways S and C flavors could be in a three scoop serving?

Answer 1

1a) for each gift there are 3 choices (number of children) ; therefore number of ways =3⁸ =6561

b) number of ways at most 2 children have all the gifts =N(selecting 2 child from 3 and for each gift there are 2 choices )=³C₂ *2⁸ =3*256=768

as above 768 cases will include 3 replicate  cases where all 8 gifts goes to 1 children

therefore number of ways to distribute so that every child receives at least one gift =6561-768+3=5796

2)

a)number of ways =only one flavour+two flavours and 1 flavour has 2 scoop while other 1 scoop+three flavours =⁵C₁+⁵C₂*2 +⁵C₃ =5+10*2+10=35

b)number of ways =5 (for 2 scoop are S and C while 3rd one can be any of 5 flavours)