Counting with restrictions

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.

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 $2$ oranges and $1$ apple?

I approached this problem by using ${7+4-1}\choose{7}$ = $120$. I used this because the first 3 spots are already filled by the oranges and apple. Would this be right way to do this problem?

• It would indeed. You are looking for the number of $4-$ tuples of non-negative numbers that add to $7$. – lulu Nov 19 '17 at 21:45
• @lulu just as an add on to understand it a little better. Say you had 24 fruits to choose. Say you had to choose 7 apples, 2 oranges, 1 pear and 1 banana. Would I just do (13+4-1 choose 13)? – Safder Aree Nov 19 '17 at 21:52
• The same reasoning would apply, so yes. – lulu Nov 19 '17 at 21:53
• @lulu Does it matter if I changed the wording to atleast 7,2,1,1? – Safder Aree Nov 19 '17 at 22:35
• Well, that's how I read the condition the first time. What else could it mean? It can't mean "exactly" as those numbers do not add to $24$. – lulu Nov 19 '17 at 22:38