Question

Suppose you are given 3 cards randomly from a well-shuffled deck of 52 cards. How many different hands of 3 cards can you have?

Answer 1

There 52 ways to choose the first card, 51 ways to choose the second card, and 50 ways to choose the third card, for a total of 52*51*50=132600 ways, when the order does matter. Here, the order doesn't matter, so we need to treat the different permutations of the same three elements as identical, i.e., we need to divide 132600 by the number of permutations of 3 elements, which is 3!=6. We get 132600/6=22100 ways.

Thus, there are 22100 different hands of 3 cards you can have.

[We can directly find the answer by the following formula: 52C3 = 52!/(3!49!) = 22100]