Permutation and Combination - Introduction and Examples (Part - 2) | Quantitative Aptitude for Competitive Examinations - Banking Exams PDF Download

Counting Rules

Multiplication

Suppose one starts his journey from place X and has to reach place Z via a different place Y. For Y, there are three means of transport - bus, train and aeroplane - from X. From Y, the aeroplane service is not available for Z. Only either by a bus or by a train can one reach Z from Y. Also, there is no direct bus or train service fro Z from X. We want to know the maximum possible no. of ways by which one can reach Z from X.

Sol:

= 3  × 2
= 6
If a work A can be done in m ways and another work B can be done in n ways and C is the final work which is done only when both A and B are done, then the no. of ways of doing the final work ?
Sol :
C = m  × n
C = 3 × 2 = 6

Addition rule

Suppose there are 42 men and 16 women in a party. Each man shakes his hand only with all the men and each woman shakes her hand only with all the women. We have to find the maximum no. of handsakes that have taken place at the party.

From each group of two persons we have one handshake.

Case 1 : Total no. of handshakes among the group of 42 men

⁴²C₂ = 42!/2! (42-2)! = 21 × 41 = 861

Case 2 : Total no. of  handshakes among the group of 16 women

¹⁶C₂ = 16!/2! (16-2)! = 8 × 15 = 120

so maximum no. of handshakes = 861 + 120 = 981.

Problems and Solutions

Ques 1. How many numbers of five digits can be formed with the digits 1,3,5 7 and 9 no digit being repeated ?Sol :
the no. of digits = 5

Required no. = ⁵P₅ = 5! = 120

Ques 2. How many three-digit numbers can be formed by using the digits in 735621, if repetition is not allowed ?

Sol:

ⁿP_r = n! / (n-r)!

⁶P₃ =  6! / (6-3)!

⁶P₃ = 6!/3!

⁶P₃ = 120

Ques 3. Find the number of different words that can be formed from the word 'SUCCESS'.

Sol : No. of Permutation = n! / p! × q!, where p = of one type , q = ( of another type ).

No. of Permutation = 7!/ 3! × 2!

No. of Permutation = 420

Ques 4. How many different 5 - digit numbers can be formed by using the digits of the number 713628459 ?
Sol :

ⁿP_r = n! / (n-r)!

⁹P₅ = 9! / (9-5)!

⁹P₅ = 9! / 4!

⁹P₅ = 15,120 

Ques 5. How many numbers of five digits can be formed with the digits 0,2,4,6 and 8 ?

Sol:

Ques 6. How many numbers of five digits can be formed with the digits 0,1, 2, 3, 4, 6 and 8 ?

Sol :

Here nothing has been said about the repetition of digits. So , it is understood that repetition of digits is not allowed .

Ques 7. How many even numbers of three digits can be formed with the digits 0,1, 2, 3, 4, 5 and 6 ?

Sol :

Total of such numbers = 5 × 5 v 3 = 75
req no. = 30+75 = 105

Ques 8. A round table conference is to be held between delegates of 15 companies. In how many ways can they be seated if delegates from two MNCs may wish to sit together ?
Sol :
Since delegates from two multinational companies will sit together, so considering these two delegates as one unit, there will be 13 + 1 = 14 delegates who can be arranged in a circular table in 14! ways.
The two delegates from the MNCs can be arranged among themselves in 2! ways.
Using the product rule, the required no. of ways = 14!×2!

Ques 9. A person has 12 friends out of which 7 are relatives. In how many ways can he invite 6 friends such that at least 4 of them are relatives ?
Sol:

1. Factorial Notation:

Let n be a positive integer. Then, factorial n, denoted n! is defined as:

n! = n(n – 1)(n – 2) … 3.2.1.

Examples:

1. We define 0! = 1.
2. 4! = (4 x 3 x 2 x 1) = 24.
3. 5! = (5 x 4 x 3 x 2 x 1) = 120.

 2. Permutations:

The different arrangements of a given number of things by taking some or all at a time, are called permutations.

Examples:

1. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
2. All permutations made with the letters a, b, c taking all at a time are:
   ( abc, acb, bac, bca, cab, cba)

3. Number of Permutations:

Number of all permutations of n things, taken r at a time, is given by:

ⁿP_r = n(n – 1)(n – 2) … (n – r + 1) = 

Examples:

1. ⁶P₂ = (6 x 5) = 30.
2. ⁷P₃ = (7 x 6 x 5) = 210.
3. Cor. number of all permutations of n things, taken all at a time = n!.

4. An Important Result:

If there are n subjects of which p₁ are alike of one kind; p₂ are alike of another kind;p₃ are alike of third kind and so on and p_r are alike of r^th kind, 
such that (p₁ + p₂ + … p_r) = n.

Then, number of permutations of these n objects is = 

5. Combinations:

Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.

Examples:

1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.Note: AB and BA represent the same selection.
2. All the combinations formed by a, b, c taking ab, bc, ca.
3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
4. Various groups of 2 out of four persons A, B, C, D are:

   AB, AC, AD, BC, BD, CD.

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

6. Number of Combinations:

The number of all combinations of n things, taken r at a time is:

Note:

1. ⁿC_n = 1 and ⁿC₀ = 1.
2. ⁿC_r = ⁿC_{(n – r)}

Examples: