Permutations and Combinations

A permutation is an arrangement and combination is a selection. Obviously, there would be more permutations possible than combinations for a given set of elements.

In a permutation the order of arrangement is important, where as in a combination , order is not important since we are concerned only with what elements we are selecting.

Fundamental Principle of Counting
If a work can be done in either m or n ways, the total number of ways of doing it is m + n .

If a work can has to be done by both m and n ways, the total ways of doing it is mn ways.

If a work has to be done under a restriction, then No. of Ways = Total ways without restriction - no. of ways under opposite restriction

Permutations
No. of permutations = nPr = n! / (n-r)!

Permutations of repeated objects
if an object is repeated p times and another object is repeated q times, then

No. of Permutations = nPr/ p!!q!

Circular Permutations
No. of Permutations = (n-1)! ;     If clockwise and anticlockwise are different

No. of Permutations = 1/2 (n-1) ! ; if clockwise and anti clockwise are identical

Combinations
No. of Combinations = nCr= n! / r!(n-r)!

Number of selections from n objects of zero or more objects = 2n

without zero = 2n - 1

No. of selections from n identical objects without selecting zero = n

with zero = n + 1

No. of selections from p identical, q identical and r distinct objects :

if all and none can be selected = (p+1)(q+1)2r .

If none cannot be selected = (p+1)(q+1)2r - 1

if all and none cannot be selected = (p+1)(q+1)2r - 2

Number of Selections of r things from n different things when each can be repeated unlimited times

= n+r-1Cr

Distributions
No. of ways of distributing n different things between 2 people, one reciving p things and other receiving q things , where p + q = n (all distributed)

= nCpxn-pCq== n! / p!q!

similarly for three people, = '''n! / p!q!r! ;   '''p+q+r = n

Number of ways to distribute combination of two objects (mxn) among n people equally is

= (mn)!/ (m!)n

No. of ways to divide (mxn) objects into n bundles

= (mn)!/ (m!)nn!

No. of ways to divide n identical objects among r people

= n+r-1Cr-1

Tips And Tricks
1) nCr = nCn-r

2) nCr + nCr-1 = n+1Cr

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