Combination formula and facts

What is combination:
Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.

Examples:

1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.

Note: AB and BA represent the same selection.

2. All the combinations formed by a, b, c taking ab, bc, ca.

3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.

4. Various groups of 2 out of four persons A, B, C, D are:

AB, AC, AD, BC, BD, CD.

5. Note that ab ba are two different permutations but they represent the same combination.

6. Number of Combinations:
The number of all combinations of n things, taken r at a time is:
\begin{aligned}
^n C_r = \frac{n!}{(r!)(n-r)!} \\
= \frac{\text{n(n-1)(n-2).... to r factors}}{r!} \\
\text{Note :} ^n C_n = 1 &, ^n C_0 = 1
\end{aligned}