A pizza shop offers nine toppings. No topping is used more than once. What is the probability that the toppings are pepperoni, onions, olives, and mushrooms?
There are 2^9 number of subsets or 512
This includes the nullset, which corresponds to a pizza with no items.
{pepperoni, onions, olives, and mushrooms} is one of those subsets.
So the Prob(pepperoni, onions, olives, and mushrooms) = 1/512
To find the probability, we need to determine how many different combinations of four toppings can be made with nine toppings available.
Since the order of the toppings does not matter, we can use the concept of combinations. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items we want to choose.
In this case, we have 9 toppings to choose from, and we want to choose 4 toppings:
9C4 = 9! / (4!(9-4)!) = 9! / (4!5!) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 126
So, there are 126 different combinations of four toppings that can be made from the nine available.
Now, since we are interested in the specific combination of pepperoni, onions, olives, and mushrooms, we need to determine how many different combinations we can make with these four toppings.
Out of the four toppings we want, there is only one way to arrange them.
Therefore, the probability of getting the specific combination of pepperoni, onions, olives, and mushrooms is 1 out of the total number of combinations we calculated:
Probability = 1/126 ≈ 0.0079 or 0.79%
To find the probability, we need to know the total number of possible combinations of toppings and the number of favorable outcomes.
1. Total number of possible combinations of toppings:
Since no topping is used more than once, we can count the combinations using the concept of permutations. In this case, we have 9 choices for the first topping, 8 choices for the second topping, 7 choices for the third topping, and 6 choices for the fourth topping. So, the total number of possible combinations is:
9 * 8 * 7 * 6 = 3,024
2. Number of favorable outcomes:
Since we want the toppings to be pepperoni, onions, olives, and mushrooms, we have only one favorable outcome.
3. Calculating the probability:
The probability is given by the number of favorable outcomes divided by the total number of possible combinations:
Probability = Number of favorable outcomes / Total number of possible combinations
Probability = 1 / 3,024
Therefore, the probability that the toppings are pepperoni, onions, olives, and mushrooms is 1/3,024.