A certain club consists of 5 men and 6 women. a) How many ways are there to form a committee of 3 people and b) How many ways are there to form a committee consisting of 3 men and 4 women.
a) you would simply choose 3 of the 11
= C(11,3) = 165
b) 3 men of 5, 4 women of 6
= C(5,3) x C(6,4)
= 10x15 = 150
c d
a) To form a committee of 3 people, we can choose from the pool of 5 men and 6 women. We don't discriminate here, so both men and women are eligible for selection. Let's call this "genderly inclusive committee formation."
Now, how many ways are there to pick 3 people from a group of 11? It's like when you're trying to decide which of your friends to bring with you to a party. You can't decide, so you put their names in a hat and draw 3 lucky winners.
Using this method, there are:
11! / (3! x (11-3)!) = 165 different ways to form a committee.
But let's not forget the possibility of a magic show! In magic, you can create illusions and make things disappear. So, if you decide to make one of the men or women vanish, you can subtract the number of committees with a missing person from the total.
Hope that clears things up, or at least amused you a little! Now, let's tackle part b.
b) We're now looking for a committee consisting of 3 men and 4 women. This time we need to be gender-specific, keeping our focus on the equality between men and women.
To form a committee with 3 men and 4 women, we choose from the pool of 5 men and 6 women, again using the "weightlifting hat method" to ensure fairness.
Using the same logic as before, we have:
5! / (3! x (5-3)!) x 6! / (4! x (6-4)!) = 60 different ways to form such a committee.
Remember, laughter is always the best way to solve math problems!
a) To form a committee of 3 people from the club, we can choose 3 members out of the total 11 members (5 men + 6 women). This is a combination problem.
The number of ways to choose 3 people from 11 is given by the formula:
nCr = n! / (r! * (n-r)!)
Plugging in the values, we have:
n = 11 (total number of members)
r = 3 (number of members to choose for the committee)
nCr = 11! / (3! * (11-3)!)
= 11! / (3! * 8!)
Simplifying further:
nCr = (11 * 10 * 9 * 8!) / (3! * 1 * 8!)
= (11 * 10 * 9) / (3 * 2 * 1)
= 165
Therefore, there are 165 ways to form a committee of 3 people from the club.
b) To form a committee consisting of 3 men and 4 women, we need to choose 3 men out of the 5 men in the club, and 4 women out of the 6 women in the club. These are two separate combination problems.
To choose 3 men from 5 men:
nCr = n! / (r! * (n-r)!)
= 5! / (3! * (5-3)!)
= (5 * 4 * 3!) / (3! * 2!)
= 5 * 4 / 2
= 10
To choose 4 women from 6 women:
nCr = n! / (r! * (n-r)!)
= 6! / (4! * (6-4)!)
= (6 * 5 * 4!) / (4! * 2)
= 6 * 5 / 2
= 15
To find the total number of ways to form the committee, we multiply the two results:
Total number of ways = 10 * 15
= 150
Therefore, there are 150 ways to form a committee consisting of 3 men and 4 women from the club.
To solve these problems, we will use combinations because the order of selection does not matter.
a) To find the number of ways to form a committee of 3 people regardless of gender, we need to select 3 individuals from a total of 11 (5 men + 6 women). We can use the combination formula:
nCr = n! / r!(n-r)!
Where n represents the total number of people available, and r represents the number of people to be selected.
In this case, n = 11 and r = 3. Plugging the values into the formula, we get:
11C3 = 11! / (3!(11-3)!)
Calculating further:
11C3 = 11! / (3!8!)
Simplifying:
11C3 = (11 * 10 * 9) / (3 * 2 * 1)
11C3 = 165
Therefore, there are 165 ways to form a committee of 3 people.
b) Now, let's determine the number of ways to form a committee consisting of 3 men and 4 women. We will calculate this by finding the number of combinations for each gender separately and then multiplying them.
First, let's find the number of ways to select 3 men from the total of 5 men. Applying the combination formula:
5C3 = 5! / (3!(5-3)!)
Calculating further:
5C3 = 5! / (3!2!)
Simplifying:
5C3 = (5 * 4 * 3) / (3 * 2 * 1)
5C3 = 10
Now, let's determine the number of ways to select 4 women from the total of 6 women:
6C4 = 6! / (4!(6-4)!)
Calculating further:
6C4 = 6! / (4!2!)
Simplifying:
6C4 = (6 * 5) / (2 * 1)
6C4 = 15
Finally, to determine the total number of ways to form a committee consisting of 3 men and 4 women, we multiply the number of ways to select 3 men and 4 women:
Total ways = (Number of ways to select men) * (Number of ways to select women)
Total ways = 10 * 15
Total ways = 150
Therefore, there are 150 ways to form a committee consisting of 3 men and 4 women.