Re: A pizza may be ordered with any of eight possible toppings.
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15 Jun 2023, 04:00
OE
First, note that for both quantities we’re dealing with a combination,
because the order of toppings doesn’t matter. A pizza with mushrooms
and pepperoni is the same thing as a pizza with pepperoni and
mushrooms. Let’s figure out Quantity A first.
We have eight toppings and we’re picking three of them. That means
we have three slots to fill. There are 8 toppings that could fill the first
slot, 7 that could fill the second slot, and 6 that could fill the third, so
we have 8 × 7 × 6. Since this is a combination, we have to divide by the
factorial of the number of slots. In this case we have three slots, so we
have to divide by 3!, or 3 × 2 × 1. So our problem looks like this: 8*7*6/3*2*1
To make the multiplication easier, let’s cancel first. The 6 on
top will cancel with the 3 × 2 on the bottom, leaving us with 8*7/1
which is 56. Thus, there are 56 ways to order a three-topping pizza
with eight toppings to choose from. Now let’s look at Quantity B.
We still have eight toppings, but this time we’re picking five of them so
we have five slots to fill. There are 8 toppings that could fill the first
slot, 7 that could fill the second slot, 6 that could fill the third, 5 that
could fill the fourth, and 4 that could fill the fifth. That’s 8 × 7 × 6 × 5 ×
4, but we still have to divide by the factorial of the number of slots. We
have five slots, so that means we need to divide by 5!, or 5 × 4 × 3 × 2 ×
1. Thus we have 8*7*6*5*4/5*4*3*2*1
We definitely want to cancel first here,
rather than doing all that multiplication. The 5 on top will cancel with
the 5 on the bottom. Likewise, the 4 on top will cancel with the 4 on the
bottom. The 6 on top will cancel with the 3 × 2 on the bottom, leaving
us again with 8*7/1=56
Therefore, there are also 56 ways to
order a five-topping pizza with eight toppings to choose from. The two
quantities are equal, so the answer is (C).