## Combinations -probability formulas

### Combinations

An arrangement of objects in which the order is not important is called a combination. This is different from permutation where the order matters. For example, suppose we are arranging the letters A, B and C.

In a permutation, the arrangement ABC and ACB are different. But, in a combination, the arrangements ABC and ACB are the same because the order is not important.

$$C(n,r)= \ ^nC_r= \frac {n!}{(n-r)!r!}$$

### Example:

In how many ways can a coach choose three swimmers from among five swimmers?

### Solution:

There are 5 swimmers to be taken 3 at a time.

Using the formula:

$$C(n,r)= \frac {n!}{(n-r)!r!}$$
$$C(5,3)= \frac {5!}{(5-3)!3!}$$
$$C(5,3)= \frac{5\times 4\times 3\times 2\times 1}{2\times 1\times 3\times 2\times 1 }$$
$$C(5,3)= 10$$

Using the formula:

### Factorial formula:

$$n!= n \cdot (n − 1) \cdot (n − 2)\cdots 3 \cdot 2 \cdot 1$$