Five boys and seven girls in a history class will go on a field trip during the Easter vacation. They make a committee of three members to prepare for the trip. What is the probability that this committee will consist of three boys? What is the probability that the committee will have two boys and one girl?

number of committes without restriction

= C(12,3) =220

number of committess with three boys
= C(5,3) = 10
prob( 3boys) = 10/220 = 1/22

number of committees with 2 boys, 1 girs
= C(5,2) x C(7,1)
= 70

prob (2boys, 1 girl ) = 70/220 = 7/22

Well, let's get the probability party started!

First, we need to find the total number of ways to choose a committee. Since we have 5 boys and 7 girls, the total number of students is 12. So, we'll have to choose 3 students out of those 12.

Now, to find the probability of a committee consisting of three boys, we need to find the total number of ways to choose 3 boys out of the 5 available boys and divide it by the total number of ways to choose 3 students out of the 12. The probability is:

(Number of ways to choose 3 boys) / (Number of ways to choose 3 students)

= (5 choose 3) / (12 choose 3)

= (10) / (220)

= 1/22

So, the probability of having a committee consisting of three boys is 1/22.

Now, let's move on to the second part of the question. We want to find the probability of having a committee with two boys and one girl. Similarly, we'll find the total number of ways to choose 2 boys and 1 girl out of the available students.

(Number of ways to choose 2 boys and 1 girl) / (Number of ways to choose 3 students)

= (5 choose 2) * (7 choose 1) / (12 choose 3)

= (10) * (7) / (220)

= 70/220

= 7/22

So, the probability of having a committee with two boys and one girl is 7/22.

Remember, these probabilities are just for fun and can't guarantee whether the committee will be the funniest or not!

To find the probability of each case, we need to know the total number of possible committees that can be formed.

Total number of possible committees = Total number of ways to choose 3 out of 12 students

To calculate this, we can use the combination formula:

nCr = n! / (r!(n-r)!),

where n is the total number of items and r is the number of items being chosen.

So, the total number of possible committees is:

Total number of possible committees = 12C3 = 12! / (3!(12-3)!) = 12! / (3!9!) = (12 * 11 * 10) / (3 * 2 * 1) = 220.

Now let's find the probability for each case:

1. Probability of the committee consisting of three boys:

To form a committee of three boys, we need to choose 3 out of 5 boys. So the number of ways to choose 3 boys out of 5 is:

Number of ways to choose 3 boys out of 5 = 5C3 = 5! / (3!(5-3)!) = 5! / (3!2!) = (5 * 4) / (2 * 1) = 10.

So, the probability of the committee consisting of three boys is:

Probability of three boys = Number of ways to choose 3 boys / Total number of possible committees = 10 / 220 = 1 / 22.

2. Probability of the committee consisting of two boys and one girl:

To form this committee, we need to choose 2 boys out of 5 and 1 girl out of 7. So the number of ways to choose 2 boys out of 5 and 1 girl out of 7 is:

Number of ways to choose 2 boys out of 5 and 1 girl out of 7 = (5C2) * (7C1) = (5! / (2!(5-2)!)) * (7! / (1!(7-1)!)) = (5! * 7!) / (2! * (5-2)! * 1! * (7-1)!) = (5 * 4 * 7) / (2 * 1) = 70.

So, the probability of the committee consisting of two boys and one girl is:

Probability of two boys and one girl = Number of ways to choose 2 boys and 1 girl / Total number of possible committees = 70 / 220 = 7 / 22.

Therefore, the probability that the committee will consist of three boys is 1/22, and the probability that the committee will have two boys and one girl is 7/22.

To find the probability of forming a committee consisting of three boys, we need to determine the total number of ways to select a committee and the number of ways to select a committee with three boys.

The total number of ways to form a committee of three members out of twelve students (five boys and seven girls) can be calculated using combinations. The formula for combinations is given by "nCr = n! / (r! * (n - r)!)", where n is the total number of students and r is the number of students selected.

Therefore, the total number of ways to select a committee is:

12C3 = 12! / (3! * (12 - 3)!) = (12 * 11 * 10) / (3 * 2 * 1) = 220.

Now, let's calculate the number of ways to form a committee consisting of three boys. Since there are five boys, this can be calculated using combinations as well:

5C3 = 5! / (3! * (5 - 3)!) = (5 * 4 * 3) / (3 * 2 * 1) = 10.

Therefore, there are 10 ways to form a committee consisting of three boys.

To find the probability, we divide the number of favorable outcomes (committee with three boys) by the total number of possible outcomes (total number of committees):

Probability of forming a committee with three boys = Number of ways to form a committee with three boys / Total number of ways to form a committee.

Probability of forming a committee with three boys = 10 / 220 = 1 / 22.

Hence, the probability of forming a committee consisting of three boys is 1/22.

Similarly, to find the probability of forming a committee with two boys and one girl, we calculate the number of ways to form such a committee using combinations.

The number of ways to form a committee with two boys and one girl can be calculated by selecting two boys from five and one girl from seven:

5C2 * 7C1 = (5! / (2! * (5 - 2)!)) * (7! / (1! * (7 - 1)!)) = (10 * 7) = 70.

Therefore, there are 70 ways to form a committee with two boys and one girl.

Probability of forming a committee with two boys and one girl = Number of ways to form a committee with two boys and one girl / Total number of ways to form a committee.

Probability of forming a committee with two boys and one girl = 70 / 220 = 7 / 22.

Hence, the probability of forming a committee consisting of two boys and one girl is 7/22.