Step by step
The question
Question: A team has 7 seniors and 5 juniors.
4 of them are picked at random.
What is the probability that all 4 are seniors?
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
12 C 4: the top 4 numbers from 12 down, over 4 factorial.
Before we multiply, we make the numbers smaller.
12 and 4 both divide by 4: 3.
3 and 3 both divide by 3: 1.
10 and 2 both divide by 2: 5.
What is left is 11 times 5 times 9.
11 times 5 is 55.
55 times 9 is 495.
So there are 495 ways.
3 Count the ways we want
7 C 4: the top 4 numbers from 7 down, over 4 factorial.
Before we multiply, we make the numbers smaller.
4 and 4 both divide by 4: 1.
6 and 3 both divide by 3: 2.
2 and 2 both divide by 2: 1.
What is left: 7 times 5.
That is 35 ways.
4 Write the fraction
35 ways we want, out of 495.
Careful!
The top and the bottom must count the same way: both are groups, with C.
5 Simplify
Both 35 and 495 can be divided by 5.
That makes it simpler.
So it is 7/99.
6 Check
Let's check another way: pick one at a time.
The first: 7 out of 12.
Then one fewer of each.
Multiplied, that is 7/99.
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.