# Ten lollipops are to be distri

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

Answer:

The number of ways of arranging identicalballs in distincturns 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 ofways 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 waysof 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), thenumber of ways of distributing 10 among 4 children is the number ofinteger ()solutions to the equation,

.(First distribute 1 lollipop to each child)

Which is .

Case 2: When all lollipops have different colors, use therecursion,

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

Thus,

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

The number of ways is