# Local grocery store just received a large shipment of apples, oranges, pears, and bananas—more than $100$ pieces each.

Your local grocery store just received a large shipment of apples, oranges, pears, and bananas—more than $$100$$ pieces each. You are shopping at the store and will purchase your fruit for the week.

$$1$$. How many ways can you select $$10$$ pieces of fruit from your store’s supply of apples, oranges, pears, and bananas?

Is it like there are more than $$400$$ fruits in total, so is it $$\binom{400+10-1}{10-1}$$

$$2$$. How many ways can you select $$10$$ pieces of fruit from your store’s supply of apples, oranges, pears, and bananas if you need at least one piece of each kind of fruit?

• No. This is a stars and bars problem. Commented Dec 6, 2018 at 16:21

As you can see we can arrange it in $$\frac {13!} {10!.3!}$$ = $$13 \choose 3$$ ways.
For the second part of the question, we need to pre-allocate 1 fruit each to all brackets, so you will have 4 of the 10 fruits already allocated, for the remaining 6 we can use the same method and find that we can choose in $$6+3 \choose 3$$ = $$9 \choose 3$$ ways