Question

From a group of 4 women and 6 men, a committee of 4 members will be elected. a) How many committees can you choose?
b) How many committees can be elected if there should be 3 men?
c) How many committees can be elected if there should be at least one woman?
d) If you want to choose a directive, that is, a president, vice president, treasurer and secretary, in how many ways can you choose?
e) If the whole group wants to sit down, all in a row, in how many different ways can they be done?
f) If the whole group wants to sit down, all in a row, in how many different ways can they be done, if to take if Pepito and Pepita want to be together?
g) If the whole group wants to sit, all in a row. How many different ways can be done if women are together and men are together?

Answer 1

(b)

3 Men can be selected from 6 Men in:

ways

1 Woman can be selected from 4 Women in

ways

So,

Number of committees = 20 X 4 =80

So,

Answer is:

80

(c)

Total number of committees is given by:

4 Members can be selected from 10 Members in:

ways

Number of committees without any Woman is given by:

4 Men can be selected from 6 Men in:

So,

Number of Committees with at least 1 Woman = 210 - 15 = 195

So,

Answer is:

195

(d)

President can be selected in 10 ways

Vice president can be selected in 9 ways

Treasurer can be selected in 8 ways

Secretary can be selected in 7 ways

So,

Total number of ways = 10 X 9 X 8 X 7 = 5040

So,

Answer is:

5040

(e)

Number of ways = 10! = 3628800

So,

Answer is:

3,628,800

(f)

Consider Pepito and Popita as one person.

Number of ways of arranging 9 persons = 9! = 362,880

Pepito and Pepita can be arranged in 2ways.

So,

Number of ways = 362,880 X 2 = 725,760

So,

Answer is:

725,760

(g)

Consider all Men as1 person. All Women as1 person.

There 2 persons can be arranged in 2! = 2 ways

6 Men can be arranged in 6! = 720 ways

4 Women can be arranged in 4! = 24 ways

So

Total number of ways = 2 X 720 X 24 = 34,560 ways

So,

Answer is:

34,560