# Theory on Combinations and its Applications

Combinations

Each of the different groups or selection which can be made by taking some or all of a number of given things or objects at a time is called a combinations.

Combinations (represented by nCr):
The symbol nCr denotes the number of combinations of n different things taken r at a time. The letter C stands for combination.

nCr = Number of combinations (selection) of n things taken r at a time.
Thus,5C2 will denote the number of selections of 5 different things taken 2 at a time.

nCr = $$\frac{{{\text{n}}!}}{{{\text{r}}!\left( {{\text{n}} - {\text{r}}} \right)!}}$$   where n ≥ r ( n is greater than or equal to r).
Thus, 5C2 = $$\frac{{5!}}{{2!\left( {5 - 2} \right)!}} = \frac{{5!}}{{2!3!}}$$

Typical situations where Combination (selection) is used

Selection of people for a team, a party, a job,, an office, and so on (e.g. selection of cricket team of 11 from 16 members).
Selection of objects (like letters, hats, pants, shirts) from amongst another larger set available foe selection.
In other words, any selection in which the order of selection holds no impotence is counted by using combinations.

Difference between a permutation and Combination

In a combination only selection is made whereas in a permutation selection as well as arrangement is taken into consideration.
In combination, the ordering of the selected objects is immaterial whereas in permutation, the ordering is essential. For example, 1,2 and 2,1 are same combination but different as permutation.
Practically to find the permutations of n different items or objects taken r at time, we first select r items from n items and then arrange them; so, usually the number of permutations exceeds the number of combinations.
Each combination corresponds to many permutations. For example, the six permutations 123, 132, 231, 213, 321, 312 correspond to same combination (1, 2, 3).

The relationship between Permutation and combination

\eqalign{ & ^{\text{n}}{{\text{P}}_{\text{r}}} = \frac{{{\text{n}}!}}{{\left( {{\text{n}} - {\text{r}}} \right)!}}\,.\,.\,.\,.\,.\,.\,\left( 1 \right) \cr & ^{\text{n}}{{\text{C}}_{\text{r}}} = \frac{{{\text{n}}!}}{{{\text{r}}!\left( {{\text{n}} - {\text{r}}} \right)!}}\,.\,.\,.\,.\,.\,.\,\left( 2 \right) \cr}
From equations (1) and (2), we get:
nPr = r! × nCr
nPr = nCr × rPr
The permutation or arrangement of r things out of n is nothing but the selection of r things out of n followed by the arrangement of the r selected things amongst themselves.