there are 8 women and 6 men in a department. how many ways can a committee of 4 people be selected if there must be at least two women on the committee?
8C2*6C2 + 8C3*6 + 8C4 = 826
To determine the number of ways a committee of 4 people can be selected with at least 2 women, we can use combinatorics.
First, let's consider the scenario where all 4 committee members are women:
There are 8 women in the department, so we have 8 ways to choose the first woman and 7 ways to choose the second woman (as order doesn't matter). For the remaining two members, we can choose any 2 people from the remaining 14 (8 women and 6 men). This can be calculated using the combination formula, denoted as nCr:
8C2 = 8! / (2! * (8 - 2)!) = 28 ways
Next, let's consider the scenario where 3 committee members are women and 1 is a man:
We have 8 ways to choose the first woman, 7 ways to choose the second woman, and 6 ways to choose the third woman. For the remaining member, we can choose 1 person from the remaining 6 men. This can be calculated as:
8 * 7 * 6 * 6 = 1,008 ways
Finally, let's consider the scenario where 2 committee members are women and 2 are men:
We have 8 ways to choose the first woman and 7 ways to choose the second woman. For the remaining two members, we can choose 2 people from the remaining 6 men:
8 * 7 * 6C2 = 8 * 7 * (6! / (2! * (6 - 2)!)) = 420 ways
To find the total number of ways to form the committee with at least 2 women, we sum up the three scenarios:
28 + 1,008 + 420 = 1,456 ways
Therefore, there are 1,456 ways to select a committee of 4 people with at least 2 women.
To find the number of ways to form a committee of 4 people with at least two women, we need to consider different cases.
Case 1: Selecting 2 women and 2 men
To form a committee with 2 women and 2 men, we need to select 2 women out of the 8 available and 2 men out of the 6 available.
Number of ways to select 2 women = C(8, 2) = 28 (using combinations formula)
Number of ways to select 2 men = C(6, 2) = 15
So, the number of ways to form a committee with 2 women and 2 men = 28 * 15 = 420.
Case 2: Selecting 3 women and 1 man
To form a committee with 3 women and 1 man, we need to select 3 women out of the 8 available and 1 man out of the 6 available.
Number of ways to select 3 women = C(8, 3) = 56
Number of ways to select 1 man = C(6, 1) = 6
So, the number of ways to form a committee with 3 women and 1 man = 56 * 6 = 336.
Case 3: Selecting 4 women
To form a committee with 4 women, we need to select 4 women out of the 8 available.
Number of ways to select 4 women = C(8, 4) = 70
Now, we can sum up the number of ways from each case to get the total number of ways to form a committee with at least two women.
Total number of ways = 420 + 336 + 70 = 826.
Therefore, there are 826 ways to select a committee of 4 people if there must be at least two women on the committee.