# Elitmus Permutations And Combinations Questions & Answers

**1.Five-digit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the greatest and smallest numbers that can be formed?**

A) 19800

B) 41976

C) 32976

D) None of these

**Correct Answer : C**

**Explaination :**

Greatest five digit number : 43210

Smallest five digit number : 10234

Difference = 43210-10234= 32976.

**2.A man has 9 friends: 4 boys and 5 girls. In how many ways can he invite them, if there have to be 3 exactly girls in the invitees?**

A) 320

B) 160

C) 80

D) 200

**Correct Answer : B**

**Explaination :**

3 Girls can be selected out of 5 girls in 5C3 ways.

Since number of boys to be invited is not given, hence out of 4 boys, he can invite them (2)4 ways.

Hence required number of ways is = 5C3 x (2)4 = 160.

**3.A, B, C, D are four towns, any three of which are non-collinear. Then, the number of ways to construct three roads each joining a pair of towns so that the roads do not form a triangle is?**

A) 7

B) 8

C) 9

D) 24

**Correct Answer : D**

**Explaination :**

To construct 2 roads, three towns can be selected out of 4 in 4 x 3 x 2 = 24 ways.

Now, if the third road goes from the third town to the first town, a triangle is formed and if it goes to the fourth town, a triangle is not formed.

So there are 24 ways to form a triangle and 24 ways of avoiding a triangle.

**4.A five digit number is formed using digits 1, 3, 5, 7 and 9 without repeating any one of them. What is the sum of all such possible numbers?**

A) 6666600

B) 6666660

C) 6666666

D) None of these

**Correct Answer : A**

**Explaination :**

Keeping one digit in fixed position, other four can be arranged in 4! ways= 24 ways. Thus each of the 5 digits will occur in each of the five place 4! times.

Hence the sum of digits in each position is 24(1 + 3 + 5 + 7 + 9) = 600.

So, the sum of all numbers = 600(1+10+100+1000+10000) = 6666600.

**5.Ten points are marked on a straight line and 11 points are marked on another straight line. How many triangles can be constructed with vertices from among the above points?**

A) 495

B) 550

C) 1045

D) 2475

**Correct Answer : C**

**Explaination :**

Required number of triangles formed

10C2 x 11 + 11C2 x 10= 45 x 11 + 55 x 10 = 1045.

**6.In a chess competition involving some boys and girls of a school, every student had to play exactly one game with the every other student. It was found that in 45 games both the players were girls and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is?**

A) 200

B) 216

C) 235

D) 256

**Correct Answer : A**

**Explaination :**

Let there be m boys and n girls.

Then nC2 = 45 » n(n – 1)=90» n =10

mC2 = 190 » m(m – 1) = 380 » m=20

Number of games played between one boy and one girl

= 10C1 x 20C1= 10 x 20=200

**7.An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, …. 9 such that the first digit of the code is non-zero. The code, handwritten on a slip, can however potentially create confusion, when read upside down- for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?**

A) 80

B) 78

C) 71

D) 69

**Correct Answer : D**

**Explaination :**

The available digits are 0, 1, 2 …………… 9. The first digit can be chosen in 9 ways ( 0 not acceptable ), the second digit can be accepted in 9 ways ( digits repetition not allowed). Thus the code can be made in 9 x 9=81 ways.

Now, there are only 4 digits which can create confusion 1, 6, 8, 9. The same can be given in the following ways.

Total number of ways confusion can arise = 4 x 3=12

Thus, required answer= 81-12=69.

**8.How many four-letter computer passwords can be formed using only the symmetric letters. (no repetition allowed) (Symmetric letters :- A, H, I, M, O, T, U, V, W, X AND Z)**

A) 7920

B) 330

C) 14640

D) 419430

**Correct Answer : A**

**Explaination :**

Ist place of the four letter password can be filled in 11 ways.

IInd place of four letter password can be filled in 10 ways.

IIIrd place of four letter password can be filled in 9 ways.

IVth place of four letter password can be filled in 8 ways.

Hence, required number of ways= 11 x 10 x 9 x 8=7920 ways.

**9.How many three-letter computer password can be formed with at least one symmetric letter ? (Symmetric letters: A, H, I, M, O, T, U, V, W, X and Z)**

A) 990

B) 2730

C) 12870

D) 1560000

**Correct Answer : C**

**Explaination :**

Three letter password from 26 letters can be selected in 26 x 25 x 24 ways. Three letter password from 15 asymmetric letters can be selected in 15 x 14 x 13 ways.

Hence, three letter password with at least one symmetric letter can be made in (26 x 25 x 24)-(15 x 14 x 13)=12870 ways.

**10.Ten points are marked on a straight line and 11 points are marked on another straight line. How many triangles can be constructed with vertices from among the above points?**

A) 495

B) 550

C) 1045

D) 2475

**Correct Answer : C**

**Explaination :**

Required number of triangles formed

10C2 x 11 + 11C2 x 10= 45 x 11 + 55 x 10 = 1045