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 ⁿC_r):
The symbol ⁿC_r denotes the number of combinations of n different things taken r at a time. The letter C stands for combination.

ⁿC_r = 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.

ⁿC_r = $$\frac{{{\text{n}}!}}{{{\text{r}}!\left( {{\text{n}} - {\text{r}}} \right)!}}$$   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:
ⁿP_r = r! × ⁿC_r
ⁿP_r = ⁿC_r × ^rP_r
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.

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