# ln how manyways can we distribute $7$ apples and $6$ oranges among $4$ children so that each child gets at least one apple.

In how many ways can we distribute $7$ apples and $6$ oranges among $4$ children so that each child gets at least one apple?

I think this can be solved by using permutations because the word distribute (arrange) is given, if the word select (e.g., select a group of ..) was there I was to use combinations. Also permutations takes into consideration the order of arrangement. Am I right? Can some one explain how to solve the above problem?

• Find the number of ways to distribute the apples, using Stars and Bars. Find the number of ways to distribute the oranges, using Stars and Bars. Multiply. – André Nicolas Feb 20 '14 at 4:39
• @AndréNicolas stars and bars???? – techno Feb 20 '14 at 4:50
• Please see this Wikipedia article. – André Nicolas Feb 20 '14 at 4:53
• @AndréNicolas okay,thanks :) – techno Feb 20 '14 at 4:55
• anyone any other answer? – techno Feb 20 '14 at 8:13

The strategy mentioned by Andre Nicolas is also called as Balls in Urns Principle.

Suppose you have k distinguishable urns and n indistinguishable balls,there are $\dbinom{n+k-1}{k}$ ways of arranging the balls in urns.

Also,$\dbinom{n+k-1}{n}$ = $\dbinom{n+k-1}{k-1}$,which you can easily verify.

In the given question,there are 4 distinguishable children,3 indistinguishable apples and 6 indistinguishable oranges.Since every child has to have a apple,you have no choice over 4 apples.

Hence,there are $\dbinom{3+4-1}{4-1}$ ways of choosing apples and $\dbinom{6+4-1}{4-1}$ ways of choosing oranges.

Using the multiplication principle,there are $\dbinom{3+4-1}{4-1}$*$\dbinom{6+4-1}{4-1}$ of doing them together.

Hint: Give each child their required 1 apple. Thus all you actually have to do is distribute 6 oranges and the remaining 3 apples among the 4 children. Consider each fruit separately, then multiply.

• okay,so let me consider the case of distributing 6 oranges to 4 children.I understand that permutations does takes into consideration the order of distribution.So cannot use i use 6P3 here right?,Please clear my doubts here – techno Feb 20 '14 at 4:47