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:
- 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
- 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
