# Combinatorics and Probability

Combinatorics can be very useful in calculating possible and favorable outcomes, especially for a large number of events.

#### Examples

1. A group of 10 people sit on a bench. What is the probability that two of the people decided in advance to sit together?

Possible outcomes:

Since the order matters and the elements are not repeated, there are permutations.

Favorable outcomes:

If the two people sitting together are considered as a single person there will be 9!, however, there can only be two possible forms for each of them, left or right, so there is 2 · 9!.

2.Five cards are drawn randomly from a standard deck consisting of 52 cards. Find the probability of picking:

4 aces.

Since the order does not matter and there are no repeated elements, there are combinations.

4 aces and 1 king.

3 fives and 2 jacks.

A 9, 10, jack, queen and king (in no particular order).

3 of any suit and 2 from another.

There are four ways of choosing the first suit and three ways of choosing to the second suit.

At least one ace.