A committee of 4 must be chosen from 3 women and 4 men.calculate
1. In how many ways the committee can be chosen.
2. In how many ways 2 men and 2 women can be chosen
3. Probability that the committee consists of 2 men and 2 women
4. Probability that the committee. consists of atleast 2 women.
1. no restrictions, number of committees
= C(7,4) = 35
2. 2 men and 2 women,
number is C(4,2) x C(3,2)
= 6(3)
= 18
3. same set-up as #2, you try it
4. at least 2 women
---> 2 women, 2 men + 3women, 1 man
= ...
let me know what you got
Truee have succeeded
1. To calculate the number of ways the committee can be chosen, we use the concept of combinations. The formula for calculating combinations is nCr, where n is the total number of items and r is the number of items chosen.
In this case, we have 7 people in total (3 women + 4 men) and we need to choose a committee of 4. Therefore, the number of ways the committee can be chosen is:
7C4 = (7!)/(4!(7-4)!) = (7!)/(4!3!) = (7x6x5)/(3x2x1) = 35
So, there are 35 ways in which the committee can be chosen.
2. To calculate the number of ways 2 men and 2 women can be chosen, we need to calculate the combinations separately for men and women, and then multiply them.
For men, we need to choose 2 out of 4, so the number of ways is:
4C2 = (4!)/(2!(4-2)!) = (4!)/(2!2!) = (4x3)/(2x1) = 6
For women, we need to choose 2 out of 3, so the number of ways is:
3C2 = (3!)/(2!(3-2)!) = (3!)/(2!1!) = (3x2)/(2x1) = 3
To calculate the total number of ways, we multiply these two results together:
6 x 3 = 18
So, there are 18 ways to choose 2 men and 2 women for the committee.
3. To calculate the probability that the committee consists of 2 men and 2 women, we need to divide the number of ways to choose 2 men and 2 women (18) by the total number of ways to choose a committee (35):
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = 18/35 ≈ 0.5143
Therefore, the probability that the committee consists of 2 men and 2 women is approximately 0.5143.
4. To calculate the probability that the committee consists of at least 2 women, we need to consider all the possible combinations of women that can be chosen and divide it by the total number of ways to choose a committee.
The number of ways to choose 2 women out of 3 is:
3C2 = (3!)/(2!(3-2)!) = (3!)/(2!1!) = (3x2)/(2x1) = 3
The number of ways to choose 3 women out of 3 is:
3C3 = (3!)/(3!(3-3)!) = (3!)/(3!0!) = 1
The total number of ways to choose a committee is 35, as calculated in the first question.
To calculate the probability, we add the number of favorable outcomes (3 + 1 = 4) and divide it by the total number of ways:
Probability = 4/35 ≈ 0.1143
Therefore, the probability that the committee consists of at least 2 women is approximately 0.1143.
To solve this problem, we can use the concept of combinations.
1. To calculate the number of ways a committee of 4 can be chosen, we can use the combination formula. The formula for combinations is C(n, k) = n! / (k!(n-k)!), where n is the total number of items and k is the number of items to be chosen. In this case, n = 7 (3 women and 4 men) and k = 4 (committee size). Therefore, the number of ways the committee can be chosen is C(7, 4) = 7! / (4!(7-4)!) = (7*6*5*4!) / (4*3*2*1*(3*2*1)) = 35. So, there are 35 ways the committee can be chosen.
2. To calculate the number of ways 2 men and 2 women can be chosen, we need to multiply the number of ways to choose 2 men from 4 men and the number of ways to choose 2 women from 3 women. Using the combination formula, the number of ways to choose 2 men from 4 men is C(4, 2) = 4! / (2!(4-2)!) = 6, and the number of ways to choose 2 women from 3 women is C(3, 2) = 3! / (2!(3-2)!) = 3. Multiplying these together gives us 6 * 3 = 18. So, there are 18 ways to choose 2 men and 2 women.
3. The probability that the committee consists of 2 men and 2 women can be calculated by dividing the number of ways to choose 2 men and 2 women (which we found to be 18) by the total number of ways to choose a committee of 4 people (which we found to be 35). So, the probability is 18/35, which can also be expressed in decimal form as approximately 0.514.
4. To calculate the probability that the committee consists of at least 2 women, we need to consider the number of ways to choose only women (2 women and 2 men) and the number of ways to choose all women (4 women and 0 men).
The number of ways to choose 2 women and 2 men is the same as the one we found in question 2, which is 18.
To calculate the number of ways to choose all women, we simply choose 4 women from the 3 available. Using the combination formula, this gives us C(3, 4) = 3! / (4!(3-4)!) = 0. So, there are 0 ways to choose all women.
The total number of ways to choose at least 2 women is the sum of these two possibilities, which is 18 + 0 = 18.
Therefore, the probability that the committee consists of at least 2 women is 18/35, which is approximately 0.514.