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

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