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Permutation & Combination

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PERMUTATIONS AND COMBINATIONS

What are permutations?

Permutations are defined as the different arrangements possible by taking one or more of a number of articles. It is an arrangement, without replacement. An example of permutation is the different arrangements, which can be used taking the letters of the word MAT

MAT, ATM, TAM, AMT, TMA, MTA i.e., 6 ways, thus we can say that the number of permutations possible, taking all the three letters of the word MAT, is 6.

The number of permutations possible taking n different things r at a time is denoted by nPr

What is a combination?

A combination is defined as a group or a selection that is made taking one or more of a number of articles, irrespective of order. It can also be defined as a selection of objects taken from a set of articles without replacement. An example is the number of combinations possible taking two letters of the word MAT.

AT, AM, MT, i.e. three combinations. We did not take TA, MA, TM, because in a group of two persons it does not matter if it is you or me, or me and you. Thus if it is A and T or T and A it makes no difference.

The number of combinations possible taking n different things r at a time is denoted by nCr

NOTE: if there are m ways of doing a thing and, if, for each of these m ways there are n ways of doing a second thing then the total number of ways in which these two things can be done is mn.

TYPES OF PERMUTATIONS

  1. The number of permutations of n different objects taken r at a time, without repetition, is given as

    2.  

    nPr = n!/(n - r)!

  2. The number of permutations of n different objects taken r at a time, with repetition, is given as

    3.  

    P = nr

  3. The number of permutations of n different objects taken all at a time, without repetition, is given as

    4.  

    nPn = n!

  4. The number of permutations of n different objects taken all at a time, without repetition, and when a1 objects are alike of one type and a2 objects are alike of the second type a3 of the third and so on, is given as

    5.  

    nPr = n!/ (a1! a2! a3!… an!)

  5. If the articles are to be arranged around a circle then the number of ways of arranging, if there is a distinction between clockwise and counterclockwise arrangements, are given as

    6.  

    P = (n-1)!

  6. If the articles are to be arranged around a circle then the number of ways of arranging, if there is no distinction between clockwise and counterclockwise arrangements, are given as

 

P = (n-1)!/2

 

TYPES OF COMBINATIONS

  1. The number of combinations of n different objects taken r at a time, is given as

    2.  

    nCr = n!/(n - r)!r!

  2. The number of combinations of n different objects taken all at a time, is given as

    3.  

    nCr = n!/(n - n)!r! = 0

    Thus, the number of combinations of n different objects taken all at a time, is o.

  3. The number of combinations of n different objects taken r at a time, when p particular things never occur is

    4.  

    C = (n-p)Cr

  4. The number of combinations of n different objects taken r at a time, when p particular things always occur is

 

C = (n-p)Cr-p

 

PROPERTIES OF COMBINATIONS

  1. nCr = nCn-r

  2. If nCr1 = nCr2 then either r1 = r2 , or, r1 + r2 = n

  3. nCr + nCr-1 = n+1Cr

  4. nC0 + nC1+ nC2 + nC3+…… nCn = 2n1

  5. nC0 + nC2 + nC4 +….. = nC1+ nC3+ nC5 +………. = 2n-1

  6. 2n+1C0 +2n+1C1 +2n+1C2 +2n+1C3 +….+2n+1Cn = 22n

GROUPING

If we have to divide m+n distinct objects into two groups containing m and n things respectively then the number of ways is

 

A = (m+ n)!/m!n!

If m = n, then the groups are equal and the number of different ways now are,

 

A = 2m!/m!m!2!

Now if there are 2m things to be divided equally between two persons then then the number of ways are

 

A = 2m!/m!m!

Similarly for 3m, 4m and so on.

For dividing m + n + p different things in three groups each containing m, n, and p objects respectively, the number of ways in which the grouping can be done is,

 

A = (m+ n+ p)!/m!n!p!

NOTE: If 25 different objects are to be distributed among 5 different persons then, the number of ways it can be done is 25!/5!5!5!5!5! or 25!/(5!)5 but if if these objects are to form five different groups then the result will be 25!/(5!)54!.

SELECTION

The number of ways in which one or more objects can be selected from n different objects is

 

2n -1

If the selection is to be made of n different objects, and when a1 objects are alike of one type and a2 objects are alike of the second type a3 of the third and so on, is given as

 

2n(a1+1)(a2+1)(a3+1)… -1

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