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In daily life, we use permutations and combinations a lot. For example, we want to go from Newyork to New Jersey and return by a different mode of transport, out of 16 cricket probables, there are a number of ways of choosing final eleven. There are a number of ways of choosing preliminary and main subjects for civil services exam, there are many ways guests can be seated on a dining table and so on. We will try to learn the basics of Permutations and Combinations. We have covered the same in the pre algebra as well.

### Permutations and Combinations Formula

Permutation Formula is^{n}p_{r}= n!/n-r!

Where n = Total number of objects

r = The number of ways to be selected.

Combination Formula is^{n}p_{r}= n!/n-r! r!

Where n = Total number of objects

r = The number of ways to be selected.

Permutation Formula is

Where n = Total number of objects

r = The number of ways to be selected.

Combination Formula is

Where n = Total number of objects

r = The number of ways to be selected.

1) Permutation mean arrangement.

2) Combination mean selection.

3) Fundamental principle of Counting:

a) Multiplication Rule: If a work is done only when all of a number of work are done then number of ways of doing that work is equal to the product of number of way of doing separate work.

b) Addition rule: If a work is done only when any one of a number of work is done , then number of way of doing that work is equal to sum of number of work of way of doing separate work.

4) n! = 1.2.3………………..n.

0! =1

5) Number of permutation of n different things taken r at a time is denoted by n!/n-r!

a) Number of permutation of n different things = n!

b) Number of permutation of n things, out of which p are alike and are of one type, q are alike and are of second type and rest are all different n!/p!q!

Permutation:

Each of the different arrangements which can be made by taking some or all of a number of given things or object at a time is called a permutation. In permutation order of appearance of things is taken into account. The following examples are taken from algebra problems.

Each of the different arrangements which can be made by taking some or all of a number of given things or object at a time is called a permutation. In permutation order of appearance of things is taken into account. The following examples are taken from algebra problems.

Combination:

Each of the different groups or selections which can be made by taking some or all of a number of given things or object at a time is called a combination. In combination order of appearance of things is not taken into account.the following algebra answer examples explain this better.

We know that the formula for arranging n items taking r at a time is given by

P(n, r) = $\frac{n!}{(n-r)!}$

= n x (n - 1)...(n - r + 1)

Here n = 10 and r = 3

The required permutation = P(10, 3) = $\frac{10!}{(10-3)!}$ = $\frac{10!}{7!}$ = $\frac{3628800}{5040}$ = 720 ways

The number of ways 10students can be arranged in 3 seats = 210

We know that the formula for arranging n items taking r at a time is given by

P(n, r) = $\frac{n!}{(n-r)!}$

= n x (n-1)...(n - r + 1)

Here there are 4 letters in the given word. So n = 5.

Here we have to find the permutations taking 4 at a time. So r = 4

So required permutation is P(5, 4) = 5 x 4 x 3 x 2 = 24

Therefore the permutations that can made with the letters of the word “ JANET” taking 4 at a time = 24

We know that the formula for selecting n items taking r at a time is given by

C(n,r) = $\frac{n!}{[(n-r)!r!]}$

= [n x (n-1)...(n-r+1)]/[1 x 2 x ...r]

Here n =15 and r = 4

The required combination = C(15, 4) = $\frac{15!}{[(15 - 4)!4!]}$ = $\frac{15!}{11!}$ x 4! = $\frac{1307674368000}{(39916800 \times 24)}$ = 1365

The number of ways selecting 15 candidates for 4 vacancies = 1365

Below you could see permutations and combinations practice problems