There are 15 chocolate bars and 10 children. In how many ways can the chocolate

Question

There are 15 chocolate bars and 10 children. In how many ways can the chocolate bars be distributed to the children, in each of the following scenarios? 57. The chocolate bars are fungible (interchangeable) (a) (b) The chocolate bars are fungible, and each child must receive at least one. Hint: First give each child a chocolate bar, and then decide what to do with the rest. The chocolate bars are not fungible (it matters which particular bar goes where). (c) The chocolate bars are not fungible, and each child must receive at least one. Hint: The strategy suggested in (b) does not apply. Instead, consider randomly giving the chocolate bars to the children, and apply inclusion exclusion (d)

Explanation / Answer

a)

15 chocolate bar , 10 children

x1 + x2+ .. x10 = 15

here xi >= 0

number of solution = (15 +10 -1) C (10-1) = 24C9

b)

x1 +x2+ + x10 = 15

here xi > 0

which is similar to

y1 + y2+ ...y10 = 5 where yi > 0

number of solution = (5 +10 - 1)C9 = 14C9

c)

now chocolate are distinct

hence

each chocolate has 10 options

hence total number of ways = 10^15

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