Counting 4-4-4-4-7

TJ Henderson emailed to ask how to count this common cribbage hand. Here's a breakdown of how it scores:

First count 15s. You need the 7 to be part of any 15 (as the remaining cards are all even and so can't make an odd total). You can make 15 with 7 plus any two 4s. So, how many different pairs of 4s can you pick from the four 4s?

Here's one way to think about it. With only two cards, there's only one pair. If you add another card, it can pair with either of the existing ones, which gives you two new pairs, so there are 3 ways to pick 2 from 3. If you add another card, it can pair with each of the three existing cards, giving you three more new pairs, so there are 6 ways to pick 2 from 4. (How many ways are there to pick 2 from 5?)

That makes 6 15s, for 12 points.

Next count pairs. There are 6 pairs of 4s, as we just calculated, so that scores another 12 points. Three of a kind is known in cribbage as a 'pair royal' (6 points), and four of a kind is called 'double pair royal' (12 points).

Then count runs. There are none.

Finally count flushes and nobs. There are neither.

Total the scores. This hand scores 24 points!

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