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#1 2015-01-18 04:40:20

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Combinations with repetitions problem

I am stuck at a combination problem. I do not know how to logically organize the given items.

The question is:

A coffee shop sells 30 different flavours of coffee and it offers 3 different sizes. In how many ways can we order a dozen two-flavoured coffees if any two of them in one order must differ at least by the flavour or by the size?


I tried to do the following:

X = flavour 1 => 30 flavours


Y = flavour 2 => 30 flavours

S = size => 3 sizes

Now I have to find a way to first select 3 items from the sets above and then 11 more distinct ways to select those 3 items.

So the first one I have the choice of all 30 flavours for both X and Y and any size. So I can select the first one 30*30*3 ways.

Next one I can select either with a different flavour for X or Y or a different size. So the two ways are: 30*29*3 or 30*30*2

If I follow this logic through it gets into too many different cases and its hard to keep track of. Is there a better way to do it?

Last edited by careless25 (2015-01-18 08:25:25)

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#2 2015-01-18 05:15:17

ElainaVW
Member
Registered: 2013-04-29
Posts: 580

Re: Combinations with repetitions problem

Hello,

A few examples would help me not make any interpretation errors.

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#3 2015-01-18 05:22:12

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Combinations with repetitions problem

One example of 12 coffees would be:

1. (x1, y1, s1)
2. (x1, y1, s2)
3. (x1, y1, s3)
4. (x2, y1, s1)
5. (x2, y1, s2)
6. (x2, y1, s3)
7. (x3, y1, s1)
8. (x3, y1, s2)
9. (x3, y1, s3)
10. (x4, y1, s1)
11. (x4, y1, s2)
12. (x4, y1, s3)

4 to 6 are equivalent to (x1, y2, s1), (x1, y2, s2), and (x1, y2, s3). So do not double count these.
Same goes for 7-9 and 10-12.

Last edited by careless25 (2015-01-18 05:26:40)

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#4 2015-01-18 05:38:58

ElainaVW
Member
Registered: 2013-04-29
Posts: 580

Re: Combinations with repetitions problem

I am working on it and have sent it off to Bobbym.

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#5 2015-01-18 06:09:57

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,048

Re: Combinations with repetitions problem

So, flavors xi and yi are the same, right?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#6 2015-01-18 06:24:47

ElainaVW
Member
Registered: 2013-04-29
Posts: 580

Re: Combinations with repetitions problem

Are these two the same?

(x4, y1, s2)
(x4, y2, s3)

Must both flavors be different too?

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#7 2015-01-18 08:17:37

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Combinations with repetitions problem

anon: Yes both flavors xi and yi are the same.

Elaina: No those two are not the same, they differ since both the yi's and si's are different. No both flavours can be same as long as the size is different.

Last edited by careless25 (2015-01-18 08:32:54)

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#8 2015-01-18 08:34:50

ElainaVW
Member
Registered: 2013-04-29
Posts: 580

Re: Combinations with repetitions problem

Are these two considered different?

(x4, y1, s2)
(x4, y2, s2)

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#9 2015-01-18 09:04:00

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Combinations with repetitions problem

Yes those are also different.

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#10 2015-01-18 09:10:13

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,048

Re: Combinations with repetitions problem

Okay, so I think it would go like this:


The number of ways of choosing a combination of flavors is
, because those are combination with replacement. Then we also multiply by 3, for the sizes. So, there are
combinations. From those we need to choose 12 different ones, which we can do in
ways.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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