Step by step
The question
Question: A club has 6 boys and 4 girls.
3 of them are picked at random.
What is the probability that all 3 are boys?
The probability is the number of ways we want, over all the ways.
P stands for probability.
1 Order: yes or no
Does order matter?
No.
Picking Ann, then Bo, is the same group as Bo, then Ann.
So we count groups: combinations.
2 Count all the ways
10 C 3: the top 3 numbers from 10 down, over 3 factorial.
Before we multiply, we make the numbers smaller.
9 and 3 both divide by 3: 3.
10 and 2 both divide by 2: 5.
What is left is 5 times 3 times 8.
5 times 3 is 15.
15 times 8 is 120.
So there are 120 ways.
3 Count the ways we want
6 C 3: the top 3 numbers from 6 down, over 3 factorial.
Before we multiply, we make the numbers smaller.
6 on top and 3 times 2 at the bottom: 6 divided by 6 is 1.
What is left: 5 times 4.
That is 20 ways.
4 Write the fraction
20 ways we want, out of 120.
Careful!
The top and the bottom must count the same way: both are groups, with C.
5 Simplify
Both 20 and 120 can be divided by 20.
That makes it simpler.
So it is 1/6.
6 Check
Let's check another way: pick one at a time.
The first: 6 out of 10.
Then one fewer of each.
Multiplied, that is 1/6.
The same.
So the answer is right!
Let's look back.
First: order: yes or no.
Then: count all the ways.
Then: count the ways we want.
Then: write the fraction.
Then: simplify.
Last: check.