# Thread: Combinatorics exercise (numbers, digits...)

1. ## Combinatorics exercise (numbers, digits...)

There are four cards. In each there's one of these numbers written: 1, 2, 3, 4 (every used once). Three digit numbers are made with the cards, putting the cars variously. How many three digit numbers can be made up, that can be divided by three?

My first question: is there a formula that'll help me to calculate this, or do I need to write all the possible three digit numbers and choose ones that can be divided by 3?

2. ## Re: Combinatorics exercise (numbers, digits...)

I don't know a formula that would give the final answer. However, I recommend first finding out which (or, rather, how many) subsets of three cards are allowed and then using the standard formula for the number of permutations of each subset.

4x3x2x1=24?

4. ## Re: Combinatorics exercise (numbers, digits...)

I have a different answer. Do you want to use the suggestion I provided above?

5. ## Re: Combinatorics exercise (numbers, digits...)

Originally Posted by emakarov
I have a different answer. Do you want to use the suggestion I provided above?
I'm sorry, I don't come from an English speaking country so it's hard for me to understand what 'subsets of three cards' or 'number of permutations' mean.

6. ## Re: Combinatorics exercise (numbers, digits...)

Originally Posted by Evaldas
There are four cards. In each there's one of these numbers written: 1, 2, 3, 4 (every used once). Three digit numbers are made with the cards, putting the cars variously. How many three digit numbers can be made up, that can be divided by three?
Recall that in order for a number to be divisible by three the sum of its digits must be divisible by three.
There are six ways to arrange $\{1,2,3\}$.

7. ## Re: Combinatorics exercise (numbers, digits...)

By "allowed subsets" in post #2 I meant those sets of three cards (or digits) that form numbers divisible by 3. For example, {2, 3, 4} is allowed because any number built from these thee digits is divisible by 3, but {1, 2, 4} is not allowed because numbers built from these digits are not divisible by 3.

You need to answer two questions: how many numbers can be built from a given subset of three digits, and how many 3-element subsets of digits are allowed.

8. ## Re: Combinatorics exercise (numbers, digits...)

So allowed subsets: {1, 2, 3}; {2, 3, 4}.
Amount of numbers that can be built from an allowerd subset: {1, 2, 3} 3x2x1=6; same for {2, 3, 4}?
2 3-element subsets of digits are allowed.

9. ## Re: Combinatorics exercise (numbers, digits...)

Yes, so 12 numbers can be formed.

Thank you!