# Combinatorics

Combinatorics is a stream of mathematics that concerns the study of finite discrete structures. Features of combinatorics involve:

- Counting the structures of the provided kind and size.
- To decide when a particular criteria can be fulfilled and analyzing elements of the criteria, such as combinatorial designs.
- To identify “greatest”, “smallest” or “optimal” elements, known as external combinatorics.
- Combinatorial structures that rise in an algebraic concept, or applying algebraic techniques to combinatorial problems, known as algebraic combinatorics.

### Difference Between Combination and Permutation

In English, we make use of the word “combination” without thinking if the order is important. Let’s take a simple instance.

The fruit salad is a combination of grapes, bananas, and apples. The order of fruits in the salad does not matter because it is the same fruit salad.

But, the combination of a key is 475. You need to take care of the order, since the other combinations like 457, 574, or other won’t work. Only the combination of 4 – 7 – 5 can unlock.

Hence, to be precise;

- When the order does not have much impact, it is said to be a combination.
- When the order does have an impact, it is said to be a permutation.

### Mathematical form of Permutation and Combination:

**Permutation:** The act of a arranging all the members of a set into some order or sequence, or rearranging the ordered set, is called as the process of permutation.

Mathematically Permutation is given as

k-permutation of n is:

\(^{n}P_{k} = \frac{n!}{(n-k)!}\)

**Combination:** Selection of members of set where order is disregarded.

k-combination of n is:

\(^{n}C_{k} = \frac{n!}{k!.(n-k)!}\)

Combinatorics Examples

1. Arrange the letters of the word TAMIL so that

- T is always next to L

- T and L are always together

Solution:

- Let’s consider LT as one letter and keep it together. Now we have 4 alphabets that can be arranged in a row in \(^{4}P_{4}\) = 24. (General formula, nc
_{r})

- L and T can be interchanged in their positions in 2! Ways. Therefore, total arrangements is given by 4!2! = 48.

2. Calculate the number of ways a cricket eleven can be selected out of a batch of 15 players if;

- no restriction on the selection.

- A specific player is always selected.

- A specific player is never chosen.

Solution:

- When there is no restriction on the selection. This means 15c
_{11}gives you the total number of ways.

- Since a specific player is selected always. Therefore, 15 -1, we have 14c
_{10}

- Since a specific player is never selected, we have 14c
_{11}

3. Calculate the number of committees of 5 students that can be chosen from a class of 25.

Solution:

Since we have to select 5 out of 25. Therefore,

\(^{25}C^{5}= \frac{25 \times 24 \times 23 \times 22 \times 21 }{5 \times 4 \times 3 \times 2 \times 1}\)

= 53130