Question

In how many ways can 12 indistinguishable apples and 1 orange be distributed among three children in such a way that each child gets at least one piece of fruit?

Consider the multiset {n .a,l,2,3,...,n) of size 2n. Determine the number of its n-combinations.

consider the multiset {n.a,n.b,1,2,3,...,n+1)of size 3n+1.Determine the number of its n-combinations

Answer 1

these are 3 different questions

i am doing the first one

answer 1

since each child should get at least one piece of fruit and all fruits have to be distributed, do the distribution in two stages.

stage 1 = choose one of the three children to get the orange and the other two for one apply each. there are 3 ways

stage 2 = for each outcome of the stage 1, distribute the remaining 10 apples to the three children. each child has the ability to take as many as he/she is given. this becomes problem of 10 combinations of a multiset with three types of objects (apples recieved by each child) with infinite repetition numbers ( each child can take as many as given) thus there are =

=

finally, multiply the ways from two stages = 3 *66 = 198