# ways are there to distribute $7$ (identical) apples, $6$ oranges and $7$ pears among $3$ different people with each person getting at least $1$ pear?

How many ways are there to distribute $7$ (identical) apples, $6$ oranges and $7$ pears among $3$ different people with each person getting at least $1$ pear?

Below are my workings but I am not sure if they are correct, looking for some help, thanks!

Using the inclusion/exclusion principle:

Include the number of ways to distribute them such that at most $\color\red3$ people have pears:

$$3^7\cdot\binom{3}{\color\red3}\cdot\color\red3^6=1594323$$

Exclude the number of ways to distribute them such that at most $\color\red2$ people have pears:

$$3^7\cdot\binom{3}{\color\red2}\cdot\color\red2^6=419904$$

Include the number of ways to distribute them such that at most $\color\red1$ person has pears:

$$3^7\cdot\binom{3}{\color\red1}\cdot\color\red1^6=6561$$

Hence the number of ways to distribute them such that each person has pears is:

$$1594323-419904+6561=1180980$$

• This number is far too large. You are, I think, treating the various apples (say) as distinguishable. There aren't, for example, $3^7$ ways to distribute the apples, far from it. Hint: apply Stars and Bars to each type of fruit separately. – lulu Feb 20 '18 at 19:26
• @lulu thanks, could you help me through this? – fr14 Feb 20 '18 at 19:29

It's important to recognize that the fruit (of a given type) are meant to be indistinguishable. Thus, factors like $3^7$ should not appear.

We will use Stars and Bars:

For the pears, we need the number of $3-$tuples of positive integers that add to $7$. That's $$\binom 62=15$$

For the apples, we need the number of $3-$tuples of non-negative integers that add to $7$. That's $$\binom {7+3-1}{7}=36$$

For the oranges, we need the number of $3-$tuples of non-negative integers that add to $6$. That's $$\binom {6+3-1}{6}=28$$

Finally, the answer is the product $$\boxed {15\times 36\times 28 = 15120}$$

Note: as always with combinatorics, it's always a good idea to check the arithmetic carefully.

• I see where I went wrong, thanks! – fr14 Feb 20 '18 at 19:41