How to calculate cribbage 29 hand odds

In our Cribbage Statistics page we stated, that the getting of the best cribbage hand - a 29 hand - odds is 1 to 216580. How did we calculate it.

First of all we should remember, what upon dealing a player receives 6 cards, 2 of them will be discarded to the crib later. Also we know, that there are only 4 hands that give 29 points:
5D 5C 5H JS and Cut Card 5S
5D 5C 5S JH and Cut Card 5H
5D 5H 5S JC and Cut Card 5C
5C 5H 5S JD and Cut Card 5D

So that our goal is to determine the probability that the initial six card hand contains three fives and a jack, all of different suits, but not the remaining five (which needs to be the starter card).

There is a combinatorics' formula for calculating the number of the all possible combinations of r elements selected from the set of k total elements:
C_(k,r) = k! / (r! * (k-r)!)
where k! means k factorial i.e. k! = 1*2*3*....*k;

In cribbage it means we select 6 cards from the 52-card pack

C_(52,6) = 52! /(6! * (52-6)!) = 52! / (6!*46!) = 52*51*50*49*48*47 / 720 = 20358520 - the total number of 6 card hands
The good hands: there are four 4-card hands. And - 2 free cards left to add to 6 cards totally (except for the one starter card 5 - one for each set). So exclude 5 required card from the set - we can select our free 2 cards from the set of 52-5 = 47 cards.
It will give the next formula:
4 * C_(47,2) = 4 * 47!/ (2! * (47-2)!) = 4* 47 * 46 / 2 = 4324 - the number of the good 6-card cribbage hands.

Now we have 4324 from 20358520 without a cut card.
A cut card we can pick 1 from the other 46 cards not in our hands, so the final calculation is 4324 / 20358520 / 46 = 1 / 216580

Therefore, the probability of getting a 29 hand in 6 card cribbage is 1 to 216,580 or 0.000004617

This article appeared as a result of the question of Ron Smyrski about the best cribbage hand odds - thank you, Ron.