Eleven people, 7 men and 4 women, successfully completed their study at the School of Astronautics. A team of 6 people needs to be selected from them for the next intergalactic travel. In how many ways can 6 astronauts be selected, if at least 2 women and 3 men must be in the team?

thank u!!

For at least 2 women and 3 men, the only cases would be

2W,4M = C(4,2) x C(7,4) = 6x35 = 210 , or
3W,3M = C(4,3) x C(7,3) = 4x35 = 140

for a total of 350 ways

(all possible ways would be
0W,6M
1W,5M
2W,4M
3W,3M
4W 2M )

Worked out combination of women and men

Either 3 women and 3 men or 4 men and 2 women
Then comination for the 2 Women was n = 4 no. of combos was 6 and r = 2 and for the 3 women n = 4 and r = 3 no. of combos was 4

Follow the formula n!/(n-r)! x (r!)

Same process for men. In each instance 35 combinations for Men.

Then times combination for each set which was (6 x 35) + (4 x 35) = 350

350 combinations for the team

To find the number of ways to select a team of 6 astronauts from the group of 11 (7 men and 4 women), we need to consider two cases: when exactly 2 women and 3 men are selected, and when more than 2 women and 3 men are selected.

Case 1: Exactly 2 women and 3 men are selected.
To calculate this, we need to choose 2 women out of 4 and 3 men out of 7.
The number of ways to select 2 women from 4 is given by the combination formula: C(4, 2) = 4! / (2! * (4-2)!) = 6.
Similarly, the number of ways to select 3 men from 7 is C(7, 3) = 7! / (3! * (7-3)!) = 35.
Therefore, the number of ways to select exactly 2 women and 3 men is 6 * 35 = 210.

Case 2: More than 2 women and 3 men are selected.
To calculate this, we need to consider all possible combinations of selecting 3 women (3 out of 4) and 3 men (3 out of 7).
The number of ways to select 3 women from 4 is C(4, 3) = 4! / (3! * (4-3)!) = 4.
Similarly, the number of ways to select 3 men from 7 is C(7, 3) = 7! / (3! * (7-3)!) = 35.
Therefore, the number of ways to select more than 2 women and 3 men is 4 * 35 = 140.

Now, we can find the total number of ways to select a team of 6 astronauts by adding the two cases together: 210 + 140 = 350.

Therefore, there are 350 different ways to select a team of 6 astronauts if at least 2 women and 3 men must be in the team.