# 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, ncr)
• 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 15c11 gives you the total number of ways.
• Since a specific player is selected always. Therefore, 15 -1, we have 14c10
• Since a specific player is never selected, we have 14c11

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