Question

Ten lollipops are to be distributed to four children. All lollipops of the same color are
considered identical. How many distributions are possible if: (a) all lollipops are red; (b) all
lollipops have different colors; (c) there are four red and six blue lollipops? (d) What are
the answers if each child must receive at least one lollipop?

Answer 1

The number of ways of arranging identical balls in distinct urns is the number of terms in the expansion of . This is . Or this is the number of

integer () solutions to the equation,

Ten lollipops are to be distributed to four children.

a) When all lollipops are red (indistinguishable), the number of ways of distributing 10 among 4 children is the number of integer () solutions to the equation,

Which is .

b) When all lollipops have different colors, the number of ways of distributing 10 among 4 children (here order is important) is

c)When there are four red and six blue lollipops as in part (a), multiply numbers of solutions to equations,

and . i.e.

d) If each child must receive at least one lollipop,

Case 1: When all lollipops are red (indistinguishable), the number of ways of distributing 10 among 4 children is the number of integer () solutions to the equation,

. (First distribute 1 lollipop to each child)

Which is .

Case 2: When all lollipops have different colors, use the recursion,

Let be the number of ways of distributing different lollipops to 4 children (each child must receive at least one lollipop) , then

Thus,

Case 3: When there are four red and six blue lollipops. First distribute one lollipop (either red or blue) each to 4 children then distribute the remaining 6 to 4.

The number of ways is