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
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