The most important mathematical areas for poker players are: Knowing how to calculate the odds, game theory (not covered here) and knowing how to calculate the number of possibilities for cards to come or being dealt, this is known as permutation. This is covered here with some examples. You might also want to check the wikipedia page for a more in depth mathematical view.

Number of all poker hands

Calculating the number of possibilities for all 5 card poker hands:

To draw 5 cards from a deck with 52 cards you have 52 * 51 * 50 * 49 * 48 = 311.875.200 combinations. The order in which the cards are dealt is not important, so this number must be divided by the number of combinations to receive 5 cards: 5 * 4 * 3 * 2 * 1 = 120.

311.875.200 divided by 120 equals 2.598.960, the number of possibilities to draw 5 cards from 52 card deck. This is expressed as (52 choose 5) or P(52,5) . For a more in depth mathematical information on this topic please read "5-Card Poker Hands", from Brian Alsbach.

Number of all Texas Hold'em Starting Hands

(52 choose 2) = 52 * 51 / (2 * 1) = 1.326. If you regard equal to (AK suited) or equal to (AK off suit) then there are just 169 possible starting hand combinations. Here you find a table with all hands.

Number of all Seven Card Stud Starting Hands

The result is: (52 choose 3) = 52 * 51 * 50 / (3 * 2 * 1) = 22.100 different starting hands.

Number of all Omaha Starting Hands

The result is: (52 choose 4) = 52 * 51 * 50 * 49 / (4 * 3 * 2 * 1) = 270.725 different starting hands.

Number of all Seven Card Poker Hands

The result is: (52 choose 7) = 52 * 51 * 50 * 49 * 48 * 47 * 46 / (7 * 6 * 5 * 4 * 3 * 2 * 1) = 133.784.560 hands.

Number of possible 5 Card Poker Hands out of 7 Cards

The result is: (7 choose 2) = 7 * 6 / (2 * 1) = 21 possible combinations.

Number of Possibilities to form a Hand in Omaha High or High Low

In Omaha you have to use 2 cards out of your hole cards (4 choose 2)= 4 * 3 / (2 *1) = 6 ways to pick two cards. and use three cards from the board (5 choose 3) = 5 * 4 * 3 / (3 * 2 * 1) = 10 ways to pick three cards from the board. Every combination from your two hole cards with 3 cards from the board is possible, so this two numbers are multiplied for a total of 60 possible poker hands.
In Omaha High Low this number doubles to 120 as you the same calculation applies for the low end.

Number of possible Flops for a Texas Hold'em Starting Hand

As your hand removed two cards from the deck, the numebr of flops is (50 choose 3)= 50 * 49 * 48 / (3 * 2 * 1) = 19.600

Number of possible Fifth Street for a Seven Card Stud Starting Hand

As your hand removed three cards from the deck, the numebr of flops is (49 choose 2)= 49 * 48 / (2 * 1) = 1.176