out of 7 men and 5 women, 5 members of a committee are selected. in how many ways can this be done if (a) there must be exactly 3 men (b) there must be more women than men?
exactly 3 men ---> need 2 women
number of choices
= C(7,3) x C(5,2)
= 35 x 10
= 350
more women than men
---> 5 women, 4W-1M, 3W-2M
= C(5,5) + C(5,4)xC(7,1) + C(5,3)xC(7,2)
= 1 + 35 + 10(21)
= 246
To find the number of ways to select a committee with certain conditions, we can use combinations.
(a) There must be exactly 3 men:
We need to choose 3 men from a group of 7 men and 5 women, then select 2 more members from the remaining group.
Step 1: Select 3 men from the group of 7 men.
The number of ways to select 3 men from a group of 7 is denoted by C(7, 3) which is calculated as C(7, 3) = 7! / (3! * (7 - 3)!) = 35.
Step 2: Select 2 members from the remaining group of 9 (5 women + 4 men).
The number of ways to select 2 members from a group of 9 is denoted by C(9, 2) which is calculated as C(9, 2) = 9! / (2! * (9 - 2)!) = 36.
Step 3: Multiply the two results together to get the total number of ways.
Total number of ways = C(7, 3) * C(9, 2) = 35 * 36 = 1260.
Therefore, there are 1260 different ways to select a committee with exactly 3 men.
(b) There must be more women than men:
We can analyze this condition by considering the three possibilities: 4 women and 1 man, 5 women and 0 men, or 3 women and 2 men.
1. Four women and one man:
Step 1: Select 4 women from the group of 5 women.
The number of ways to select 4 women from a group of 5 is denoted by C(5, 4) which is calculated as C(5, 4) = 5.
Step 2: Select 1 man from the group of 7 men.
The number of ways to select 1 man from a group of 7 is denoted by C(7, 1) which is calculated as C(7, 1) = 7.
Step 3: Multiply the two results together to get the total number of ways for this case.
Total number of ways = C(5, 4) * C(7, 1) = 5 * 7 = 35.
2. Five women and no men:
Step 1: Select 5 women from the group of 5 women.
The number of ways to select 5 women from a group of 5 is denoted by C(5, 5) which is calculated as C(5, 5) = 1.
Step 2: There are no men to select from, as no men are allowed.
Step 3: The total number of ways for this case is simply the number of ways to select 5 women, which is 1.
3. Three women and two men:
Step 1: Select 3 women from the group of 5 women.
The number of ways to select 3 women from a group of 5 is denoted by C(5, 3) which is calculated as C(5, 3) = 10.
Step 2: Select 2 men from the group of 7 men.
The number of ways to select 2 men from a group of 7 is denoted by C(7, 2) which is calculated as C(7, 2) = 21.
Step 3: Multiply the two results together to get the total number of ways for this case.
Total number of ways = C(5, 3) * C(7, 2) = 10 * 21 = 210.
Finally, we add up the results for all three cases to get the total number of ways:
Total number of ways = 35 + 1 + 210 = 246.
Therefore, there are 246 different ways to select a committee with more women than men.