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1. In how many ways can six different rings be worn on four fingers of one hand ?

A. 10

B. 12

C. 15

D. 16

**Answer : ****15**

Six different rings can be put in ^{6}C_{4} ways in four fingers.

^{6}C_{4} --> ^{6}C_{2} = (6 x 5) / (2 x 1) = 15

Thus, six different rings can be worn on four fingers in **15 ways**.

2. In how many ways can the letters of the word ‘LEADER’ be arranged ?

A. 72

B. 144

C. 360

D. 720

**Answer : ****360**

The word ‘LEADER’ contains 6 letters namely 1L , 2 E, 1A,1D, and 1R.

Total letters --> 6

Repeating letter --> 2 (ஃ E)

ஃ Possible ways = 6! / 2! = 6 x 5 x 4 x 3 x 2 x 1 / 2 = 360

3. In how many ways a committee, consisting of 5 men and 6 women can be formed from 8 men and 10 women?

A. 266

B. 5040

C. 11760

D. 86400

**Answer : ****11760**

From 8 men if we select 5 men then possible ways --> ^{8}C_{5}

From 10 women if we select 6 men then possible ways --> ^{10}C_{6}

Total number of ways = ^{8}C_{5} x ^{10}C_{6}

ஃ^{n}C_{r} = ^{n}C_{(n-r)}

Therefore, ^{8}C_{5} = ^{8}C_{(8-5)} = ^{8}C_{3}.

^{10}C_{6} = ^{10}C_{(10-6)} = ^{10}C_{4}.

Total number of ways = ^{8}C_{3} x ^{10}C_{4}

= (8 x 7 x 6) / (3 x 2 x 1) x (10 x 9 x 8 x 7) / (4 x 3 x 2 x 1)

=11760

4. Kate, Demi, Madona, Sharon, Britney and Nicole decided to lunch together in a restaurant. The waiter led them to a round table with six chairs. How many different ways can they seat?

A. 60

B. 120

C. 240

D. 480

**Answer : ****120**

For circular seat arrangement, the possible ways = ( n-1) ! where n is total number of chairs/things.

Required answer = (6-1)! = 5! = 120

**Another way:**

Note that on a round table ABCDEF and BCDEFA is the same.

The first person can sit on any one of the seats. Now, for the second person there are 5 options, for the third person there are 4 options, for the forth person there are 3 options, for the fifth person there are 2 options and for the last person there is just one option.

Thus, total different possible seating arrangements are = 5 * 4 * 3 * 2 * 1 = 120

5. In how many ways can 21 books on English and 19 books on Hindi be placed in a row on a shelf so that two books on Hindi may not be together ?

A. 3990

B. 1540

C. 1995

D. 3672

**Answer : ****1540**

Let E - Position of English book

Let H - Position of Hindi book.

In order to keep Hindi books that are never together, we must place all these books as:

**H E H E H E H .... H E H**

Since there are 21 books on English, we can choose 22 places to place 19 Hindi books.

^{22}C_{19} = ^{22}C_{3} = (22 x 21 x 20)/(3 x 2 x 1) = 1540 ways.

Hence, the required number of ways = 1540.

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