Create a question that uses factorials and permutations and solve it. Be clear as to how the question uses both permutations and factorials.

EXAMPLE: Your favourite pizza place offers 8 different toppings. How many different ways can you order a pizza?

To answer this question, you need to consider multiple cases.

Case 1: 1 topping pizza

Out of the 8 toppings, choose 1 for the pizza:

Case 2: 2 toppings:

Case 3: 3 toppings:

Case 4: 4 toppings:

Case 5: 5 toppings:

Case 6: 6 toppings:

Case 7: 7 toppings:

Case 8: 8 toppings:

If you notice, this is almost the entire 8th row of Pascal's triangle:

Knowing that is great, because you could just look up Pascal's triangle rather than calculating all of the cases on a calculator.

However, that's not the best part. Take a look at each row of Pascal's triangle and see what all of the numbers (which is what we need in the 8th row) add up to. Look familiar?

The sums are not only doubling each time, they are the powers of 2. It also works very nicely that the exponent is the row number!

Therefore, if we need to add the 8th row of Pascal's triangle, we get: 28 = 256. However, we don't want the first number (which would represent the case where no toppings are chosen) so we subtract 1.

Therefore,

n(S) =

n(S) = 28 - 1

n(S) = 255

So, what is the probability that a random pizza has pepperoni?

n(S) = 255

A' is all pizzas without pepperoni.

n(A') =

n(A') = 27 - 1

n(A') = 127

So,

n(A) = 255 - 127

n(A) = 128

Therefore,

P(A) =

P(A) = 50.20%