Permutations and Combinations
Permutations and Combinations PDF Notes, Important Questions And Synopsis
Permutations And Combinations PDF Notes, Important Questions and Synopsis
SYNOPSIS
 Fundamental Principle of Counting
 Permutation is the number of ways to arrange things.
Eg: Password is 2045
(order matters)  It is denoted by P(n, r) and given by
P(n, r) =, where 0 ≤ r ≤ n
n → number of things to choose from
r → number of things we choose
! → factorial
 Combination is the number of ways to choose things.Eg: A cake contains chocolates, biscuits, oranges and cookies.
(Order does not matter)  It is denoted by C(n, r) and given by
C(n, r) =, where 0 ≤ r ≤
n → number of things to choose from
r → number of things we choose
! → factorial
 Permutation is the number of ways to arrange things.
 Permutation
If n is the number of distinct things and r things are chosen at a time.
i. Permutation of objects when all are not distinct:
Permutation = , → Number of things among ‘n’ which are alike of rth type.
ii. Circular permutation
1. When clockwise and anticlockwise arrangements are different:
Number of permutations: (n  1)!
2. When clockwise and anticlockwise arrangements are the same:
Number of permutations:
iii. Permutation under restrictions
Selecting and arranging r distinct objects from n
 When ‘k’ particular things are always to be included.
Number of permutations:  When a particular thing is always to be included (k = 1).
Number of permutations:  When ‘k’ particular things are never included.
Number of permutations:  When a particular thing is never included.
Number of permutations:  When ‘l’ particular things always come together.
Number of permutations: (n  l + 1)! ⨯ l!  When ‘l’ particular things never come together.
Number of permutations: n!  (n  l + 1)! ⨯ l!
i. Permutations with repetition Number of permutations: ^{n}P_{r}= n^{r} (Repetition, order matters)
Out of ← Taking 2 at a time 
ii. Permutations without repetition
(No repetition, order matters) Out of ← Taking 2 at a time 
3.Combination


ii Combinations under restriction


i. Combination with repetition Formula: (Repetition, order does not matter) Out of _{ }← Taking 2 at a time 
ii. Combination without repetition Formula: (No repetition, order does not matter) Out of ←Taking 2 at a time 
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