What are the odds that 5 cards, drawn at random, will form a straight flush, or a full house? These poker questions are mostly combinatorics. There are 52 choose 5 or 2598960 possible hands. With this as a common denominator, we only need find the number of hands in each category.
A straight flush consists of 5 cards in sequence, all of the same suit. There are 4 suits, and the sequence can start with anything from ace to 10 inclusive. That's 4×10 = 40 possibilities.
A straight is merely an ascending sequence, with the suit unconstrained. There are 10×45 = 10240 straights, but we should really subtract the straight flushes, leaving 10200 proper straights.
A flush consists of 5 cards of the same suit. There are 4 suits, and 13 choose 5 combinations in each. Multiply this out and subtract 40, giving 5108 flushes.
If you want four of a kind, pick a rank with all four suits represented, then any other card. That's 13×48 = 624.
A full house requires 3 of one rank and 2 of another. That's 13×4 times 12×6 = 3744.
Three of a kind selects a suit, and 3 from that suit, which is 13×4. Then select any two other cards, 48 choose 2, then take away the full houses. The result is 54912.
For two pair, select two ranks, 13 choose 2, and 2 cards for each, 6×6. Then bring in one of the remaining 44 cards, giving 123552.
Finally we have a pair. Choose a rank and two cards from that rank, 13×6. Then choose 3 ranks from the remaining 12, and any of 4 cards for each. This gives 1098240.
Here is a summary of the hands, starting with the best. Remember the common denominator of 2.6 million.
Straight flush = 40
Four of a kind = 624
Full house = 3744
Flush = 5108
Straight = 10200
Three of a kind = 54912
Two pair = 123552
Pair = 1098240