Selecting a committee: There are 7 women and 5 men in a department. How many ways can a committee of 4 be selected if there must be at least 2 women?
randi k bacho yeh solutionapni pund which chara lo
3 cases :
2 women, 2 men ---> C(7,2)xC(5,2) = 21(10) = 210
3 women, 1 man ---> C(7,3)xC(5,1) = 35(5) = 175
4 women, 0 men ---> C(7,4) = 35
add up the 4 cases to get 420
or, exclude the case of no women, and one woman
C(12,4) - C(7,1)xC(5,3) - C(7,0)x(C5,4)
= 495 - 70 - 5 = 420
To calculate the number of ways a committee of 4 can be selected with at least 2 women, we can consider two scenarios:
1. Selecting 2 women and 2 men from the department:
- We can choose 2 women from the 7 women in the department in C(7, 2) ways.
- We can choose 2 men from the 5 men in the department in C(5, 2) ways.
- Therefore, the total number of ways to select 2 women and 2 men is C(7, 2) * C(5, 2).
2. Selecting 3 women and 1 man from the department:
- We can choose 3 women from the 7 women in the department in C(7, 3) ways.
- We can choose 1 man from the 5 men in the department in C(5, 1) ways.
- Therefore, the total number of ways to select 3 women and 1 man is C(7, 3) * C(5, 1).
The final answer is the sum of the two scenarios:
Total number of ways = C(7, 2) * C(5, 2) + C(7, 3) * C(5, 1)
To calculate the number of ways a committee of 4 can be selected with at least 2 women, we need to consider two cases:
Case 1: 2 women and 2 men are selected.
In this case, we need to choose 2 women out of 7 and 2 men out of 5. The number of ways to do this is calculated by the combination formula:
C(7, 2) * C(5, 2) = 21 * 10 = 210
Case 2: 3 women and 1 man are selected.
In this case, we need to choose 3 women out of 7 and 1 man out of 5. The number of ways to do this is calculated by the combination formula:
C(7, 3) * C(5, 1) = 35 * 5 = 175
Thus, the total number of ways to select a committee of 4 with at least 2 women is the sum of the two cases:
210 + 175 = 385
Therefore, there are 385 ways to select a committee of 4 from a department with 7 women and 5 men, with the requirement of having at least 2 women on the committee.