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**MelanieB****Member**- Registered: 2005-11-29
- Posts: 1

This is a bonus question from my college math assignment. Please help!

"Marbles in five colors cost $1 for 12. How many different color combintations are there for a buck?"

Basically, order doesn't matter; how many combinations (repeats are allowed) are there?

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**RickyOswaldIOW****Member**- Registered: 2005-11-18
- Posts: 212

you could have 12 marbles of all the same colour

12

or 2 colours, 11 of 1 colour and 1 of the other, 10 of 1 colour and 2 of the other..

11 : 1

10 : 2

9 : 3

8 : 4

7 : 5

6 : 6

5 : 7

4 : 8

3 : 9

2 : 10

1 : 11

Now we move on to 3 colours

10 : 1 : 1

9 : 1 : 2

9 : 2 : 1

8 : 1 : 3

8 : 3 : 1

7 : 1 : 4

7 : 4 : 1

6 : 1 : 5

6 : 5 : 1

5 : 1 : 6

5 : 6 : 1

4 : 7 : 1

4 : 1 : 7

3 : 1 : 8

3 : 8 : 1

2 : 9 : 1

2 : 1 : 9

1 : 1 : 10

1 : 10 : 1

4 colours...

9 : 1 : 1 : 1

8 : 1 : 1 : 2

8 : 1 : 2 : 1

8 : 2 : 1 : 1

7 : 1 : 1 : 3

7 : 1 : 3 : 1

7 : 3 : 1 : 1

6 : 1 : 1 : 4

6 : 1 : 4 : 1

6 : 4 : 1 : 1

5 : 1 : 1 : 5

5 : 1 : 5 : 1

5 : 5 : 1 : 1

4 : 1 : 1 : 6

4 : 1 : 6 : 1

4 : 6 : 1 : 1

3 : 1 : 1 : 7

3 : 1 : 7 : 1

3 : 7 : 1 : 1

2 : 1 : 1 : 8

2 : 1 : 8 : 1

2 : 8 : 1 : 1

1 : 1 : 1 : 9

1 : 1 : 9 : 1

1 : 9 : 1 : 1

We could continue in this way but if we look, we can see that with just one colour we have just the one combination, two colours has 12 combinations, three has 19 combinations and if we counted the rest we would see a pattern emerge. From this we can say that;

n = (c - 1) * (12 - c) + 1

where c is the number of different colours and n is the number of combinations. Thus;

if we have just one colour

n = (1 - 1) * (12 - 1) + 1

n = 0 * 11 + 1

n = 1

two colours

n = (2 - 1) * (12 - 1) + 1

n = 1 * 11 + 1

n = 12

three colours

n = (3 - 1) * (12 - 3) + 1

n = 19

four colours

n = (4 - 1) * (12 - 4) + 1

n = 25

*Last edited by rickyoswaldiow (2005-11-30 07:05:29)*

Aloha Nui means Goodbye.

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**ryos****Member**- Registered: 2005-08-04
- Posts: 394

From my College Algebra text:

Permutations: Distinct Objects with Repetition

The number of ordered arrangements of r objects chosen from n objects, in which the n objects are distinct and repetition is allowed, is n^r.

So, in your case, 5^12 = 244,140,625. That's a lot of marbles!

El que pega primero pega dos veces.

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**RickyOswaldIOW****Member**- Registered: 2005-11-18
- Posts: 212

is that 244,140,625 marbles or colour combinations? I don't think I took repeats into account

Aloha Nui means Goodbye.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Plus, you missed quite a few.

3 colours...

8:2:2

7:3:2

7:2:3

6:4:2

6:3:3

6:2:4

...and so on.

Why did the vector cross the road?

It wanted to be normal.

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**RickyOswaldIOW****Member**- Registered: 2005-11-18
- Posts: 212

>.<

I was just working it out as I went along, I've only been studying maths for 6 weeks or so

Aloha Nui means Goodbye.

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