Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: √ ∞ ≠ ≤ ≥ ≈ ⇒ ∈ Δ θ ∴ ∑ ∫ π -

Login

Username

Password

Not registered yet?

#1 2005-11-30 12:21:07

MelanieB
Novice

Offline

Tough College Perms/Combs Question

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?

#2 2005-12-01 06:04:50

RickyOswaldIOW
Full Member

Offline

Re: Tough College Perms/Combs Question

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 (2005-12-01 06:05:29)


Aloha Nui means Goodbye.

#3 2005-12-01 17:41:58

ryos
Power Member

Offline

Re: Tough College Perms/Combs Question

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.

#4 2005-12-02 04:44:36

RickyOswaldIOW
Full Member

Offline

Re: Tough College Perms/Combs Question

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


Aloha Nui means Goodbye.

#5 2005-12-02 05:06:14

mathsyperson
Moderator

Offline

Re: Tough College Perms/Combs Question

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.

#6 2005-12-02 05:20:11

RickyOswaldIOW
Full Member

Offline

Re: Tough College Perms/Combs Question

>.<

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


Aloha Nui means Goodbye.

Board footer

Powered by FluxBB