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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?

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  • $\begingroup$ It would indeed. You are looking for the number of $4-$ tuples of non-negative numbers that add to $7$. $\endgroup$ – lulu Nov 19 '17 at 21:45
  • $\begingroup$ @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)? $\endgroup$ – Safder Aree Nov 19 '17 at 21:52
  • $\begingroup$ The same reasoning would apply, so yes. $\endgroup$ – lulu Nov 19 '17 at 21:53
  • $\begingroup$ @lulu Does it matter if I changed the wording to atleast 7,2,1,1? $\endgroup$ – Safder Aree Nov 19 '17 at 22:35
  • $\begingroup$ 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 $24$. $\endgroup$ – lulu Nov 19 '17 at 22:38
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Yes, this is the number of mappings from an unlabelled 7-element set to a labelled 4 element set (i.e. the maps from a 7-element set N to a 4-element set K, up to a permutation of N), since you just need to figure out how many ways there are to pick your remaining (non-predetermined) 7 fruit.

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