Counting Example: Texas Holdem Poker.
Each player is dealt 5 cards from a deck of 52.
How many possible poker hands are there?
Use the binomial coefficient to work this out.
52 choose 5 is 2598960
What are the chances of getting a pair?
Pick a value and pick two suits e.g. king of hearts and king of spades.
13 choose one (picking one value)
4 choose two (picking two suits)
Multiply them together (AND)
Choose three of the remaining value (e.g 10, 4, 5) (12 choose 3)
Choose a suit for each (4 choose 1 and 4 choose 1 and 4 choose 1)
Multiply all of the binomial coefficients together to get the answer 1098240