Math Is Fun Forum

I was asked a question about counting in Cribbage.  I`m not sure if it`s a permutation or a combination question.  So, I looked on line for answers to permutations/combinations and found your site.  Here`s the question...

In cribbage you can count combinations of 15`s, Pairs, and Runs.  The question is around runs and double runs.   Here`s the definition of each:

`Straight Run` - Sequence of three or more consecutive cards of any suite - one point for each card in the run

`Double Run` - Two three-card or four-card straights, including one pair; e.g. 2, 3, 4, 4.  You get 2 points for the pair and 6 points for `2 different` runs of 3 - 2,3,4 and again 2,3,4 (using the other 4).

Combinations are also used which is why 3 of a kind is worth 6 points - 3 combinations of pairs, 2 points for each pair.

Now, if you have 4 cards in a single run, e.g. A, 2, 3, 4, it is worth 4 points.

My son was asked by his girlfriend, `since you can re-use the 4 in a double-run (the example above) to create two unique runs...why can`t I count these four cards (A, 2, 3, 4) as…A,2,3 for 3 points; 2, 3,4 for another 3 points; and A, 2, 3, 4 for another 4 points and thus a total of 10 points?`

First, counting this way is not allowed in Cribbage, but try as I might, I couldn`t explain why.  Second, a double run, makes sense to me - 2 distinct runs of 3 and replacing the number 4 card with its pair.  But, the comment I get back when I say you can`t count it this way is…`it`s the same thing, you`re re-using cards and so am I`.

Can anyone explain it mathematically?