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ich are not flush.
Now as respects flushes their number is very easily
determined. The number of combinations of five cards which can be formed out of the 13 cards of a suit are given by multiplying together 13, 12, 11, 10, and 9, and dividing by the product of 1, 2, 3, 4, 5; this will be found to be 1,287. Thus there are 4 times 1,287, or 5,148 possible flushes. Of these 5,108 are not sequence flushes.
The total number of ' four' hands may be considered next. The process for finding it is very simple. There are of course only 13 fours, each of which can be taken with any one of the remaining 48 cards; so that there are 13 times 48, or 624 possible four hands.
Next, to determine the number of ' full hands.' This is not difficult, but requires a little more attention. A full hand consists of a triplet and a pair. Now manifestly there are four triplets of each kind--four sets of three aces, four of three kings, and so forth (for we may take each ace from the four aces in succession, leaving in each case a different triplet of aces; and so with the other denominations). Thus, in all, 4 times 18, or 52 different triplets can be formed out of the pack of 52 cards. When one of these triplets has been formed there remain 49 cards, out of which the total number of sets of two which can be formed is obtained by multiplying 49 by 48 and dividing by two; whence we get 1,176 such combinations in all. But the total number of pairs which can be formed from among these 49 cards is much smaller. There are four twos, which (as cribbage teaches us)will give six pairs of twos; so there are six pairs of threes, six pairs of fours, and so on; or
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