Possible ways to order 7 drinks with 3 sizes with condition.

[mcjord1992]

I have a homework problem that I can't wrap my head around.

You have 7 drink requests. 3 possible sizes: sm, med, lg. How many possible ways can you order each drink assuming at least two orders are small?

At first I thought it might be:
1 = 1
2 = 1
3 = 3
4 = 3
5 = 3
6 = 3
7 = 3

Then I realized there's 6 ways you could order that assuming the first order is the sm.
Then there are 5 ways if the second order is sm.
Then 4 ways if the 3rd order is sm.

I'm getting lost in the possibilities. Can someone help me out please?

 

[HallsofIvy]

I would start by treating the case where it is the first two drinks that are small. That leaves 5 drinks each of which can be "small" medium" or "large". There are $\displaystyle 5^3$ ways to do that. Now think of those first two "small" drinks as represented by "s" and each of the other 5 drinks as represented by "o". Then the two small drinks being first would be "ssooooo". But how many different orders are the for those 7 symbols. If there were 7 distinct symbols there would be 7! different orders. But 2 of them are the same so we need to cancel the different ways that are just permutations of those 2 "s"s. There are 2! ways to do that so we need to divide by 2!. Similarly, there are 5! ways to interchange the 5 "o"s so we need to divide by 5!. That is, there are $\displaystyle \frac{7!}{2!5!}= 21$ different orders of "ssooooo".

There are 21(5^3)= 21(125)= 2625 different way to do this.

 

[pka]

I have a homework problem that I can't wrap my head around.
You have 7 drink requests. 3 possible sizes: sm, med, lg. How many possible ways can you order each drink assuming at least two orders are small?

Are we to assume tat there are seven different flavors? The question uses the word requests for drinks.
Please clarify the question.

 

[Dr.Peterson]

You have 7 drink requests. 3 possible sizes: sm, med, lg. How many possible ways can you order each drink assuming at least two orders are small?

I'm getting lost in the possibilities. Can someone help me out please?

We first have to decide exactly what the problem means. Did you quote the exact question as given to you, or is it so awkward because you have tried to restate it?

I suppose there are 7 people who each request one drink, of one size. I have to assume that we don't "have" 7 such requests already made, since we are expected to "order" them subsequently. So here's my paraphrase of what I think it must mean:

Seven distinct people are about to order drinks. Each will order one drink that is either small, medium, or large. In how many ways can these orders be made, if at least two are small?​

The "at least" part is the hard part. We can' say there are exactly 2 small and 5 other.

I would use negation: Find the total number of possible orders with no restriction, and subtract from that the number of orders that do not include at least two smalls (that is, either no smalls or exactly one small).

Can you try that? We'll need to see the details of your thinking in order to confirm, correct, or redirect your thinking.