A pizza contains three toppings randomly selected from the f

A pizza contains three toppings randomly selected from the following choices: pepperoni, mushrooms, green pepers, ham, bacon, and onions. Manuel enjoys all toppings, except bacon and onions.

What is the likelihood that pizza will contain:

a. no toppings that Manuel dislikes?
b. exactly one topping that Manuel dislikes?
c. at least one topping that manuel dislikes?

Re: A pizza contains three toppings randomly selected from t

Hello, kimmy_koo51!

A pizza contains three toppings randomly selected from the following choices:
pepperoni, mushrooms, green pepers, ham, bacon, and onions.
Manuel enjoys all toppings, except bacon and onions.

What is the likelihood that pizza will contain:

(a) no toppings that Manuel dislikes?
(b) exactly one topping that Manuel dislikes?
(c) at least one topping that manuel dislikes?

Click to expand...

There are: \(\displaystyle \:{6\choose3}\:=\:20\) possible choices of 3 toppings.

There are 4 toppings that Manuel likes (L), and 2 that he dislikes (D).

(a) We choose 3 toppings from the 4 L's: \(\displaystyle \:{4\choose3} \:=\:4\) ways.
Therefore: \(\displaystyle \,P(\text{no D's}) \:=\:\frac{4}{20}\:=\:\L\frac{1}{5}\)

(b) We choose 1 topping from the 2 D's and 2 from the 4 L's: \(\displaystyle \:{2\choose1}\cdot{4\choose2} \:=\:2\cdot6\:=\:12\) ways.
Therefore: \(\displaystyle \,P(\text{one D}) \:=\:\frac{12}{20} \:=\:\L\frac{3}{5}\)

(c) The opposite of "at least one D" is "no D's", which we found in part (a).
Therefore: \(\displaystyle \,P(\text{at least one D})\;=\;1\,-\,\frac{1}{5}\;=\;\L\frac{4}{5}\)

Manuel enjoys all toppings, except bacon and onions.

Click to expand...

Doesn't like bacon?. What kind of a goofball doesn't like bacon?.