Permutation and combination is one of the significant topic in CAT exam. These are based on the logic of combinations/cases that can be formed for the particular thing to happen. This questions helps you to solve probability questions.

**Permutation and Combination for CAT Questions:**

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**Instructions (1 – 2) :**

Directions for the next two questions: Answer the questions based on the following information.

Each of the 11 letters A, H, I, M, O, T, U, V, W, X and Z appears same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters.

**Question 1:**

How many four-letter computer passwords can be formed using only the symmetric letters (no repetition allowed)?

a) 7,920

b) 330

c) 14,640

d) 4,19,430

**Question 2:**

How many three-letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?

a) 990

b) 2,730

c) 12,870

d) 15,600

**Question 3:**

Ten straight lines, no two of which are parallel and no three of which pass through any common point, are drawn on a plane. The total number of regions (including finite and infinite regions) into which the plane would be divided by the lines is

a) 56

b) 255

c) 1024

d) not unique

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**Question 4:**

There are three cities A, B and C. Each of these cities is connected with the other two cities by at least one direct road. If a traveller wants to go from one city (origin) to another city (destination), she can do so either by traversing a road connecting the two cities directly, or by traversing two roads, the, first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to B (including those via C). Similarly there are 23 routes from B to C (including those via A). How many roads are there from A to C directly?

a) 6

b) 3

c) 5

d) 10

**Question 5:**

There are six boxes numbered 1, 2, 3, 4, 5, 6. Each box is to be filled up either with a white ball or a black ball in such a manner that at least one box contains a black ball and all the boxes containing black balls are consecutively numbered. The total number of ways in which this can be done equals.

a) 15

b) 21

c) 63

d) 64

**Answers and Solutions for Permutation and Combination for CAT Questions:**

**Solutions:**

**1)** Answer (A)

The number of ways in which this can be done is 11*10*9*8 = 7920

**2)** Answer (C)

If there are 3 symmetric letters, it can be formed in 11*10*9 ways

If there are 2 symmetric letters, it can be formed in 11C2 * 15C1 * 3! ways

If there is only 1 symmetric letter, the password can be formed in 15C2*11C1*3! ways

Total = 990+330*15+630*11 = 12870 ways

**3)** Answer (A)

If there are ‘m’ non-parallel lines, then the maximum number of regions into which the plane is divided is given by

[m(m+1)/2]+1

In this case, ‘m’ = 10

So, the number of regions into which the plane is divided is (10*11/2) + 1 = 56

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**4)** Answer (A)

The possible roads are:

A -> B Let this be x

A -> C Let this be y

B -> C Let this be z

From the information given, x + yz = 33 -> (1)

z + xy = 23 -> (2)

From the options, if y = 10, x =2 and z = 3 from (2), but it doesn’t satisfy (1)

If y = 5, x = 4 and z = 3 from (2) but they don’t satisfy (1)

A possible set of numbers for (x,y,z) are (3,6,5)

Number of roads from A -> C = 6

**5)** Answer (B)

Total ways when all 6 boxes have only black balls = 1

Total ways when 5 boxes have black balls = 2

Total ways when 4 boxes have black balls = 3

Total ways when 3 boxes have black balls = 4

Total ways when 2 boxes have black balls = 5

Total ways when only 1 box has black ball = 6

So total ways of putting a black ball such that all of them come consecutively = (1+2+3+4+5+6) = 21

We hope that this Permutation and Combination for CAT Questions is very useful to you.