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 nr

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 p1 objects are of first kind, p2 objects are of the second kind, …, pk 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:

1. If nC9 = nC8. Find nC17 .
Solution:
According to the question we have nC9 = nC8
$$\frac{n!}{9!(n-9)!}$$ = $$\frac{n!}{(n-8)!8!}$$

$$\frac{1}{9}$$ = $$\frac{1}{n-8}$$
or n – 8 = 9 or n = 17
Therefore, nC17 = 17C17 = 1
2. Find the number of permutations of the letters of the word ALLAHABAD.
Solution:
Here, there are 9 objects (letters) of which there are 4A’s, 2 L’s and rest are all different.

Therefore, the required number of arrangements is
= $$\frac{9!}{4!2!}$$ = $$\frac{5*6*7*8*9}{2}$$ = 7560

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