**Permutations and Combinations Formula:**

Nowadays we see most of the people owns a phone. In every phone, there is a security procedure so that no one else except the owner can use its data. For that, we see there is a PIN lock, Security code etc mainly of 4 digits. Suppose you want to unlock it and you don’t know its password. If there is 4 digit code then how many passwords are possible? For many such situations to solve we have to study permutations and combinations formula.

**Fundamental Principal of counting:**

If one event can occur in m different ways and another event can occur in n different ways, then the total number of occurrence of the events in the given order is m × n.

**Factorial Notation for permutations and combinations formula:**

It is the representation which denotes the multiplication of all the positive integers till that number. Thus, is the way to solve factorial.

Let a number be n such as its factorial is denoted by the notation **n!.**

Thus, **n! = 1 × 2 × 3 × 4 . . . × (n – 1) × n**

**Permutation:**

The order doesn’t matter for it. There are two permutations that are with permutation repetition and permutation without repetition.

Formula is:

\(\large Permutation=\:^{n}P_{r}=\frac{n!}{(n-1)!}\)Permutation with repetition is given by **n ^{r}**

**Some special cases: **

\(^{n}P_{0}\) = 1\(^{n}P_{r}\) = 0 when r > n

\(^{n}P_{r}\) is also denoted as P(n,r).

Where,

**n** and **r** must be non-negative integers.

**r** is the length of each permutation.

**n** is the no. of elements of the set from which elements are permuted.

**!** is the factorial operator notation.

**Remember:**

The number of permutations of n objects taken all at a time, where p_{1} objects are of first kind, p_{2 }objects are of the second kind, …, p_{k} objects are of the kth kind and rest, if any, are all different is

\(\frac{n!}{p_{1}!\:p_{2}!\: …. P_{k}!}\)

**Combination:**

The combination formula denotes the number of ways a particular of “r” elements can be obtained from a larger set of “n” distinguishable objects.

Formula is:

\(\LARGE Combination = \:^{n}C_{r} = \frac{^{n}P_{r}}{n!}\)

Written as:

\(\LARGE Combination = \:^{n}C_{r} = \frac{n!}{(n-r)!r!}\)

Where,

**n** and **r** must be non negative integers.

**r** is the length of each permutation.

**n** is the no. of elements of the set from which elements are permuted.

**!** is the factorial operator notation.

**Examples related to permutations and combinations formula:**

According to the question we have*If*Solution:^{n}C_{9}=^{n}C_{8}. Find^{n}C_{17}.

^{n}C_{9}=^{n}C_{8}\(\frac{n!}{9!(n-9)!}\) = \(\frac{n!}{(n-8)!8!}\)

\(\frac{1}{9}\) = \(\frac{1}{n-8}\)

or n – 8 = 9 or n = 17

Therefore,^{n}C_{17}=^{17}C_{17}= 1Here, there are 9 objects (letters) of which there are 4A’s, 2 L’s and rest are all different*Find the number of permutations of the letters of the word ALLAHABAD.*Solution:

*.*

Therefore, the required number of arrangements is

= \(\frac{9!}{4!2!}\) = \(\frac{5*6*7*8*9}{2}\) = 7560

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