IMPORTANT FACTS AND FORMULAE

We have always faced the problem of understanding the difference between permutation and combination.

This chapter will definitely clear the concepts of permutation and combination, the only thing you have to do is thoroughly understand the difference between the two terms and as well learn the quick tips to solve problems based on this chapter.

Permutation: The various ways of arranging a given number of things by taking some or all at a time are all called as permutations.

Permutation includes word formation, number formation, circular permutation, etc.

In permutation, objects are to be arranged in particular order. It is denoted by n P or P(n, r).

Example: Arrange the given 3 numbers 1, 2, 3 by taking two at a time.
Now these numbers can be arranged in 6 different ways: (12, 21, 13, 31, 23, 32).

Here,

12 and 21, 13 and 31 or 23 and 32 do not mean the same, because here order of numbers is important.

Example: All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).

Example: All permutations made with the letters a, b, c taking all at a time are:
(
abc, acb, bac, bca, cab, cba).

Condition 1:  Number of permutations of n things, taken r at a time is given as follows:

 nPr = n (n – 1) (n – 2) (n – 3)……. (n – r + 1) = n! (n – r)!

Condition 2:  If there are N balls and out of these B1 balls are alike, B2 balls are alike , B3 balls are alike and so on and Br are alike of rth kind, such that (B1 balls + B2 balls + B3 balls ----- Br balls) = N balls.

In such condition,

 Number of permutation of these N balls = N! (B1)! × (B2)! × (B3)! × - - - - - (Br)!

Condition 3:  If number of permutations of n objects are all taken at a time,

 then, nPr = n! = n! 0!

Important points to remember:

1. If N different objects are to be arranged, then they can be arranged in N! ways.

2. N number of objects can be arranged around a circle in (N – 1)! ways.

3. Sometimes we have to solve problems on permutation considering the condition of Repetition

Repetition: This condition is not used unless specified. (Remember)
Number of permutation of N objects taken r at a time when each selected object can be repeated any number of times is given as:

Number of permutations =
n r

4. Restricted Permutation: The number of permutations of n objects taken r at a time in which if k particular objects are:

a) Never included: (n – k)Pr ---- (k are the number of objects not included)
b) Always included:
(n – k) Cr–k x r! ---- (k is the number of objects always included)

Combination: Each of different groups or selections formed by taking some or all number of objects is called a combination.

Combination is used in different cases which include team/group/committee.

In combination, objects are selected randomly and here order of objects doesn’t matter. It is denoted by n C ror C(n, r)

Example - 1: If we have to select two girls out of 3 girls X, Y, Z, then find the number of combinations possible.

Now only two girls are to be selected and arranged. Hence, this is possible in 3 different ways: (XY, YZ, XZ,).

Here,
You cannot make a combination as XY and YX, because these combinations mean the same.

Example - 2:  Various groups of 2 out of four persons A, B, C, D are: (AB, AC, AD, BC, BD, CD).

Note that ab ba are two different permutations but they represent the same combination.

1. Number of combinations of n objects, taken r at a time is given as follows:

 nCr = n! = n(n – 1) (n – 2)…….to r factors (r!)(n – r)! r!

This example will surely clear the concept!

Hint: In the example discussed below, the confusion related to addition and multiplication of terms will also be cleared.

Example: Suppose there are 12 boys and 8 girls, and we have to select 5 volunteers for a particular task. So we have to find the number of possible selections we can make.

Total students are (12 + 8) = 20 and we have to select 5 volunteers.

 The total number of selections can be made nCr ways:
 20C5 n! (r!)(n – r)!

The question may be asked in different ways. 2 different conditions are specified below:

1) Out of 5 volunteers, 3 boys and 2 girls must be present.

2) Boys should be in majority.