SOLUTION: A class contains 8 boys and 14 girls. Two students are to be selected as student council representatives. What is the probability both are boys?

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Question 844908: A class contains 8 boys and 14 girls. Two students are to be selected as student council representatives. What is the probability both are boys?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
there are 8 boys and 14 girls.
you will be selecting 2 out of the 22 total.
the number of ways that can occur is 22C2 = 231.
the number of ways you can select 2 boys out of a total of 8 boys is 8C2 = 28
the probability that you will select 2 boys is the number of ways you can select 2 boys out of 8 divided by the number of ways you can select 2 students out of 22.
that probability would be equal to 28 / 231 = .12 rounded to 2 decimal places.

let's look at it another way.
the probability that the first pick will be a boy is 8/22.
the probability that the second pick will also be a boy is 7/21
the probability that both will be a boy is 8/22 * 7/21 which is equal to 56 / 462 which is equivalent to 28 / 231.

the decimal equivalent to this is .12 rounded to 2 decimal places.
the simplest fraction will be 4/33 since both 28 and 231 are divisible by 7.

22C2 is equal to 22! / (2! * 20!)
8C2 is equal to 8! / (2! * 6!)
the general combination equation is:
nCx = n! / (x! * (n-x)!)

the best way to see this is to use much smaller numbers so you can identify each possible combination.
assume 3 boys and 2 girls and you pick a team of 2 and you want to know the probability that you will pick 2 boys.
assuming boys are letters and girls are number, you will get the following possible combinations of 2 students.
ab
ac
a1
a2
bc
b1
b2
c1
c2
12
that's a total of 10 possible combinations of 2 students.
out of those 10 possible combinations, 3 are all boys.
they are:
ab
ac
bc
the rest are either boy and girl or all girl.

the formulas you would use in this simplified example are:

3C2 / 5C2 which results in 3/10.
3/5 * 2/4 which results in 6/20 which results in 3/10.

either method results in the same probability as it should.