5
Q:

# A Committee of 5 persons is to be formed from a group of 6 gentlemen and 4 ladies. In how many ways can this be done if the committee is to be included atleast one lady?

 A) 123 B) 113 C) 246 D) 945

Explanation:

A Committee of 5 persons is to be formed from 6 gentlemen and 4 ladies by taking.

(i) 1 lady out of 4 and 4 gentlemen out of 6

(ii) 2 ladies out of 4 and 3 gentlemen out of 6

(iii) 3 ladies out of 4 and 2 gentlemen out of 6

(iv) 4 ladies out of 4 and 1 gentlemen out of 6

In case I the number of ways = $C_{1}^{4}×C_{4}^{6}$ = 4 x 15 = 60

In case II the number of ways = $C_{2}^{4}×C_{3}^{6}$ = 6 x 20 = 120

In case III the number of ways = $C_{3}^{4}×C_{2}^{6}$ = 4 x 15 = 60

In case IV the number of ways = $C_{4}^{4}×C_{1}^{6}$ = 1 x 6 = 6

Hence, the required number of ways = 60 + 120 + 60 + 6 = 246

Q:

In how many different ways can the letters of the word 'POVERTY' be arranged ?

 A) 2520 B) 5040 C) 1260 D) None

Explanation:

The 7 letters word 'POVERTY' be arranged in $\mathbf{7}{\mathbit{P}}_{\mathbf{7}}$ ways = 7! = 5040 ways.

2 31
Q:

A decision committee of 5 members is to be formed out of 4 Actors, 3 Directors and 2 Producers. In how many ways a committee of 2 Actors, 2 Directors and 1 Producer can be formed ?

 A) 18 B) 24 C) 36 D) 32

Explanation:

Required Number of ways = = 36

8 141
Q:

How many six digit odd numbers can be formed from the digits 0, 2, 3, 5, 6, 7, 8, and 9 (repetition not allowed)?

 A) 8640 B) 720 C) 3620 D) 4512

Explanation:

Let the 6 digits of the required 6 digit number be abcdef

Then, the number to be odd number the last digit must be odd digit i.e 3, 5, 7 or 9

The first digit cannot be ‘0’ => possible digits = 3, 5, 7, 2, 6, 8

Remaining 4 places can be of 6 x 5 x 4 x 3 ways

This can be easily understood by

Therefore, required number of ways = 6 x 6 x 5 x 4 x 3 x 4 = 36 x 20 x 12 = 720 x 12

8640 ways.

3 53
Q:

In how many ways can letter of the word RAILINGS arrange so that R and S always come together?

 A) 1260 B) 2520 C) 5040 D) 1080

Explanation:

The number of ways in which the letters of the word RAILINGS can be arranged such that R & S always come together is

Count R & S as only 1 space or letter so that RS or SR can be arranged => 7! x 2!

But in the word RAILINGS, I repeated for 2 times => 7! x 2!/2! = 7! ways = 5040 ways.

3 59
Q:

On the occasion of New Year, each student of a class sends greeting cards to the others. If there are 21 students in the class, what is the total number of greeting cards exchanged by the students?

 A) 380 B) 420 C) 441 D) 400

Explanation:

Given total number of students in the class = 21

So each student will have 20 greeting cards to be send or receive (21 - 1(himself))

Therefore, the total number of greeting cards exchanged by the students = 20 x 21 = 420.

7 93
Q:

How many more words can be formed by using the letters of the given word 'CREATIVITY'?

 A) 851250 B) 751210 C) 362880 D) 907200

Explanation:

The number of letters in the given word CREATIVITY = 10

Here T & I letters are repeated

=> Number of Words that can be formed from CREATIVITY = 10!/2!x2! = 3628800/4 = 907200

6 170
Q:

If (1 × 2 × 3 × 4 ........ × n) = n!, then 15! - 14! - 13! is equal to ___?

 A) 14 × 13 × 13! B) 15 × 14 × 14! C) 14 × 12 × 12! D) 15 × 13 × 13!

Explanation:

15! - 14! - 13!

= (15 × 14 × 13!) - (14 × 13!) - (13!)

= 13! (15 × 14 - 14 - 1)

= 13! (15 × 14 - 15)

= 13! x 15 (14 - 1)

= 15 × 13 × 13!

7 237
Q:

To fill 8 vacancies there are 15 candidates of which 5 are from ST. If 3 of the vacancies are reserved for ST candidates while the rest are open to all, Find the number of ways in which the selection can be done ?

 A) 7920 B) 74841 C) 14874 D) 10213

Explanation:

ST candidates vacancies can be filled by ${}^{5}C_{3}$ ways = 10

Remaining vacancies are 5 that are to be filled by 12

=> ${}^{12}C_{5}$= (12x11x10x9x8)/(5x4x3x2x1) = 792

Total number of filling the vacancies = 10 x 792 = 7920