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# Permutations and Combinations

## Permutations And Combinations PDF Notes, Important Questions and Synopsis

SYNOPSIS

1. 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 byP(n, r) =, where 0 ≤ r ≤ nn →  number of things to choose fromr → 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 byC(n, r) =, where 0 ≤ r ≤ n →  number of things to choose fromr →  number of things we choose! →  factorial
2. 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 anti-clockwise arrangements are different:
Number of permutations: (n - 1)!
2. When clockwise and anti-clockwise arrangements are the same:
Number of permutations:
iii. Permutation under restrictions
Selecting and arranging r distinct objects from n
1.  When ‘k’ particular things are always to be included.
Number of permutations:
2. When a particular thing is always to be included (k = 1).
Number of permutations:
3. When ‘k’ particular things are never included.
Number of permutations:
4.  When a particular thing is never included.
Number of permutations:
5. When ‘l’ particular things always come together.
Number of permutations: (n - l + 1)!  l!
6. When ‘l’ particular things never come together.
Number of permutations: n! - (n - l + 1)!  l!
 i. Permutations with repetition Number of permutations: nPr= nr (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

 i Total number of combinations Ways of selecting one or more things at a time.Number of combinations: When ‘k1’ alike objects of one kind, ‘k2’ alike objects of the second kind … ‘kn’ alikeobjects of the nth kind.Number of combinations: When ‘k1’ alike objects of one kind, ‘k2’ alike objects of the second kind … ‘kn’ alikeobjects of the nth kind and rest ‘p’ elements are different.Number of combinations: ii  Combinations under restriction When ‘k’ particular things are always to be includedNumber of combinations: When a particular thing is always to be included (k = 1)Number of combinations: When ‘k’ particular things are never includedNumber of combinations: When ‘k’ particular things never come togetherNumber of combinations: 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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