A department contains 12 men and 17 women.
How many ways are there to form a committee with 6 members if it must have strictly more women than men?
It is strongly recommended that you do this on paper!
how do i solve this in the calculator?
you have to list all the different possibilities in your paper.
5women 1man
4women 2men
try different combinations of 6 members where there is always more women than men
To solve this problem using a calculator, we can use the concept of combinations and perform some calculations.
1. Firstly, determine the number of possible combinations of women and men for the committee. Since there must be strictly more women than men, we can have at most 5 men and at least 1 woman in the committee.
2. Calculate the number of combinations for each possibility:
- If we have 5 men and 1 woman, we can choose the 5 men in C(12, 5) = 792 ways and the 1 woman in C(17, 1) = 17 ways. So, the number of combinations for this case is 792 * 17 = 13,464.
- If we have 4 men and 2 women, we can choose the 4 men in C(12, 4) = 495 ways and the 2 women in C(17, 2) = 136 ways. So, the number of combinations for this case is 495 * 136 = 67,320.
- If we have 3 men and 3 women, we can choose the 3 men in C(12, 3) = 220 ways and the 3 women in C(17, 3) = 680 ways. So, the number of combinations for this case is 220 * 680 = 149,600.
- If we have 2 men and 4 women, we can choose the 2 men in C(12, 2) = 66 ways and the 4 women in C(17, 4) = 2,380 ways. So, the number of combinations for this case is 66 * 2,380 = 156,840.
- If we have 1 man and 5 women, we can choose the 1 man in C(12, 1) = 12 ways and the 5 women in C(17, 5) = 6,188 ways. So, the number of combinations for this case is 12 * 6,188 = 74,256.
3. Finally, sum up the number of combinations for each case to find the total number of ways to form the committee with strictly more women than men:
Total = 13,464 + 67,320 + 149,600 + 156,840 + 74,256
Using a calculator, you can directly calculate the final result.
To solve this problem using a calculator, you can follow these steps:
1. Calculate the total number of ways to choose a committee of 6 members from the total number of people (12 men + 17 women) using the combination formula:
nCr = n! / (r!(n - r)!)
In this problem, n = 12 men + 17 women = 29, and r = 6. So the total number of ways to choose any 6 members from the group is:
nCr = 29! / (6!(29 - 6)!)
2. Calculate the number of ways to choose a committee with more women than men. Since we must have strictly more women than men, we need to consider the cases where 1, 2, 3, 4, 5, or 6 women are chosen in the committee.
For each case, calculate the number of ways to choose the desired number of women and the remaining number of men:
- Case 1: Choose 1 woman (17 ways) and 5 men (12 ways)
- Case 2: Choose 2 women (17C2 ways) and 4 men (12C4 ways)
- Case 3: Choose 3 women (17C3 ways) and 3 men (12C3 ways)
- Case 4: Choose 4 women (17C4 ways) and 2 men (12C2 ways)
- Case 5: Choose 5 women (17C5 ways) and 1 man (12C1 ways)
- Case 6: Choose 6 women (17C6 ways) and 0 men (12C0 ways)
3. Add up the results from each case to find the total number of ways to form the committee with strictly more women than men.
It is important to note that manual calculation is recommended for this problem because it involves counting combinations, which is not available directly on most calculators.