Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have six sets: {2, 6}, {2, 6}, {1, 3}, {1, 5}, {4, 5} and {3, 4}. How many different combinations can I produce by taking one value from each set so that I produce the set {1, 2, 3, 4, 5, 6}?

A possible combination is:

S1: 2 S2: 6 S3: 1 S4: 5 S5: 4 S6: 3

and another:

S1: 6 S2: 2 S3: 1 S4: 5 S5: 4 S6: 3

share|improve this question
    
To be picky, you have only five sets since $\{2,6\} = \{2,6\}$. This eliminates some possibilities, since you can only choose a 2 or a 6 from that set, as opposed to choosing a 2 from the "first" set and a 6 from the "second" one. Your meaning is clear but the notation doesn't express what you want. –  Patrick Mar 7 '12 at 16:32

1 Answer 1

up vote 1 down vote accepted

Picky note: Use curly braces for sets: $\{2,6\}$ not $(2,6)$.

You must either choose 2 from the first $\{2,6\}$ or the second $\{2,6\}$ and then are forced to choose $6$ from the opposite pair.

Next, you must choose $3$ from either $\{1,3\}$ or $\{3,4\}$.

Case 1: Choose $3$ from $\{1,3\}$ forces you to choose $1$ from $\{1,5\}$ which forces $5$ to be chosen from $\{4,5\}$ which then forces $4$ to be chosen from $\{3,4\}$

Case 2: Choose $3$ from $\{3,4\}$ forces $4$ to be chosen from $\{4,5\}$ etc.

Thus there are 2 ways to deal with 2 and 6 and 2 ways to deal with 1,3,4,5. These options are independent, so there are a total of $4=2\cdot 2$ ways to choose.

share|improve this answer
    
Is this brute-force method the best way to go? –  John Mar 7 '12 at 16:00
    
I'm not sure there is a nice "slick" way to proceed. I think brute force is about all you can do. This sort of counting problem gets really complicated as the size of the problem increases. It's not terribly far off from a knapsack problem. –  Bill Cook Mar 7 '12 at 16:42

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.