Problem:
In a group of 6 men and 5 women, a committee of 4 people is to be selected. What is the number of ways of selecting the committee with at least 3 women such that at least one woman holds the post of either a president or a vice-president?

Solution:
To solve this problem, we can use the principle of inclusion-exclusion.

Step 1: Calculate the total number of ways to select a committee of 4 people:
To calculate the total number of ways to select a committee of 4 people, we need to choose 4 people from a total of 11 people. This can be done using combinations.

Total number of ways to choose 4 people from 11 = C(11, 4) = 330

Step 2: Calculate the number of ways to select a committee with no women:
Since we want to select a committee with at least 3 women, we need to calculate the number of ways to select a committee with no women and subtract it from the total number of ways calculated in step 1.

To select a committee with no women, we need to choose 4 people from the 6 men. This can be done using combinations.

Number of ways to choose 4 people from 6 = C(6, 4) = 15

Step 3: Calculate the number of ways to select a committee with exactly 1 woman as president or vice-president:
To calculate the number of ways to select a committee with exactly 1 woman as president or vice-president, we need to choose 1 woman for the position of president or vice-president, and then choose the remaining 3 members from the remaining 10 people (5 women and 5 men).

Number of ways to choose 1 woman for the position of president or vice-president = C(5, 1) = 5

Number of ways to choose the remaining 3 members from the remaining 10 people = C(10, 3) = 120

Number of ways to select a committee with exactly 1 woman as president or vice-president = 5 * 120 = 600

Step 4: Calculate the number of ways to select a committee with exactly 2 women as president and vice-president:
To calculate the number of ways to select a committee with exactly 2 women as president and vice-president, we need to choose 2 women for the positions of president and vice-president, and then choose the remaining 2 members from the remaining 9 people (4 women and 5 men).

Number of ways to choose 2 women for the positions of president and vice-president = C(5, 2) = 10

Number of ways to choose the remaining 2 members from the remaining 9 people = C(9, 2) = 36

Number of ways to select a committee with exactly 2 women as president and vice-president = 10 * 36 = 360

Step 5: Calculate the number of ways to select a committee with at least 3 women such that at least one woman holds the post of either a president or a vice-president:
To calculate the number of ways to select a committee with at least 3 women such that at least one woman holds the post of either a president or a vice-president,