Hi Genesis!
Strategy: 1) Notice that this problem is a permutation because the items can be selected only once, meaning that order matters. 2) Because you are selecting 4 items from a set of 12, then it is implied that you can only choose each of those 4 items without replacement. 3) Let's reason through the problem. Then, finally, 4) we will inspect our work and identify a procedure we can use on every problem like this one.
For a permutation (order matters) without replacement (each item can be selected only once), we can notice the following pattern:
- There are 12 ways we can select the first item
- There are 12-1 ways we can select the second item
- There are 12-2 ways we can select the third item
- There are 12-3 ways we can select the fourth item
So, this now looks like the familiar counting problem.
A counting problem example:
How many ways can we put on 12 shirts, 11 hats, 10 pants, and 9 scarves?
Solution: 12 * 11 * 10 * 9
Aha! That means we can solve this problem the same way:
Our solution: 12 * 11 * 10 * 9 = 11,880 possibilities, total
Now let's find out how to use this procedure for every kind of permutation-without-replacement problem.
We notice that the solution is in this form:
12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
which can be written like this:
(total number of items in the whole set) factorial
(total number - number to be selected) factorial
which can be written in math shorthand as follows:
n!
(n - r)!
where n is the total number of items in the set and r is the number you are going to select.
Other examples of permutations (order matters) without replacement:
-- How many ways can you select 4 officers from a 12-member club? (12*11*10*9)
-- How many ways can you select 3 pizza toppings from a set of 7 possible toppings? (7*6*5)
-- How many ways can you select 4 competitors for the Platinum, Gold, Silver, Bronze Awards out of a field of 20 competitors? (20*19*18*17)
Send me a message if you have any questions about this. I will be glad to help!
-- Lindsey
