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two which can be formed is obtained by multiplying 48 by 47 and dividing by 2; it is therefore 1,128, and among these 72 are pairs. There remain then 1,056 sets of two, any one of which can be combined with each of
52 triplets to give a triplet hand pure and simple. Thus, in all, there are 52 times 1,056 triplet hands, or 54,912, as before.
Next for double and single pairs.
From the whole pack of 52 cards we can form six times 13 pairs; for 6 aces can be formed, 6 pairs of deuces, 6 pairs of threes, and so forth. Thus there are in all 78 different pairs. When we have taken out any pair, there remain 50 cards. From these we must remove the two cards of the same denomination, as either or both of these must not appear in the hand to be formed. There remain 48 cards, from which we can form 72 other pairs. Each of these can be taken with any one of the 46 remaining cards, except with those two which are of the same denomination, or with 44 in all, without forming a triplet. Each of these combinations can be taken with each of the 78 pairs, giving a two-pair hand, only it is obvious that each two-pair hand will be
given twice by this arrangement. Thus the total ' number of two-pair hands is half of 78 times 72 times
44; or there are 123,552 such hands in all.
Next, as to simple pairs. We get, as before, 78 different pairs. Each of these can be taken with any set of three formed out of the 48 cards left when the other 2 of the same denomination have been removed, except the 72 times 44 (that is 3,168)pairs indicated in dealing with the last case, and the 48 triplets which can be formed out of these same 48 cards, or 3,216 sets in ail. Now the total number of sets of three cards which can be formed out of 48 is given by multiplying
48 by 47 by 46, and dividing by the product Of the numbers 1, 2, ant1 3. It is found to be 17,296. We diminish this by 3,216, getting 14,082, and find that there are in all 78 times 14,082 or 1,098,240.
The hands which remain are those which are to be estimated by the highest card in them; and their number will of course be obtained by subtracting the sum of the numbers already obtained from the total number of possible hands. We thus obtain the number 1,302,540.
Thus of the four best classes of hands, there are the following numbers:
Of flush sequences there may be. . . 40
,, fours . . . 624
,, full hands . . , 3,744
,, common flushes 5,108
· ,, common sequences . . . , 10,200
,, triplets 54,912
,, two pairs 123,552
,, pairs 1,098,240
,, other hands I . . 1,302,540
Total number of possible hands 2,598,960
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