Suppose that a department contains 8 men and 20 women. How many ways are there to form a committee with 6 members if it must have strictly more women than men?
If more women than men, possibilities are
6W, 0M
5W, 1M
4W, 2M
3W, 3M --- no longer possible
6W,0M ---> C(20,6) x C(8,0) = 38760
5W, 1M ---> C(20,5) x (8,1) = 15504(8) = 124032
4W, 2M ---> C(20,4) x C(8,2) = 4845(28) = 135660
add them up
To solve this problem, we need to determine the number of ways to select 6 members for the committee where the number of women chosen is greater than the number of men chosen.
Let's break down our approach step by step:
Step 1: Calculate the number of ways to select 6 women from a group of 20.
To solve this, we can use the concept of combinations. The number of ways to select 6 women from a group of 20 can be calculated using the combination formula, which is denoted as "nCr."
The formula for calculating combinations is:
nCr = n! / (r!(n-r)!)
In this case, we have n = 20 (the total number of women) and r = 6 (the number of women we want to select for the committee). Plugging these values into the formula, we get:
20C6 = 20! / (6!(20-6)!)
Simplifying further:
20C6 = 20! / (6!*14!)
Step 2: Calculate the number of ways to select men for the committee.
Since we need the number of women chosen to be strictly more than the number of men chosen, there must be at least 1 more woman than man. Therefore, we can consider different scenarios: 1 man and 5 women, 2 men and 4 women, 3 men and 3 women, 4 men and 2 women, or 5 men and 1 woman.
For each scenario, we calculate the number of ways to select men and women, and then sum up these values.
For 1 man and 5 women:
Number of ways to select the man = 8C1 (since we have 8 men to choose from)
Number of ways to select the women = 20C5 (calculated in step 1)
For 2 men and 4 women:
Number of ways to select the men = 8C2
Number of ways to select the women = 20C4
For 3 men and 3 women:
Number of ways to select the men = 8C3
Number of ways to select the women = 20C3
For 4 men and 2 women:
Number of ways to select the men = 8C4
Number of ways to select the women = 20C2
For 5 men and 1 woman:
Number of ways to select the men = 8C5
Number of ways to select the women = 20C1
Step 3: Sum up the values from each scenario.
To obtain the final count of all possible committee combinations, we sum up the values calculated for each scenario:
Total number of ways to form the committee = (8C1 * 20C5) + (8C2 * 20C4) + (8C3 * 20C3) + (8C4 * 20C2) + (8C5 * 20C1)
Calculating each scenario using the combination formula, we can find the total number of ways to form the committee with 6 members, where the number of women is greater than the number of men.