Question

a restaurant provides a brunch where the omelets are individually prepared. each guest is allowed to choose from 10 different ingredients. a student has been asked to form the following sets. determine the
(a) how many different types of omelets can the restaurant prepare?
(b) what is the minimum number of ingredients that must be available if the restaurant wants to advertise that it offers over 4,000 different omelets?

Explanation:

Step1: Calculate total omelet combinations

Each ingredient can be either included or excluded, so we calculate the total number of subsets of 10 ingredients, minus 1 to exclude the empty set (no ingredients is not an omelet).
$$2^{10} - 1 = 1024 - 1 = 1023$$

Step2: Solve for minimum ingredient count

We need the smallest $n$ such that $2^n - 1 > 4000$. Test values:
$2^{11}-1=2047$, $2^{12}-1=4095$. 4095 > 4000, so $n=12$.

Answer:

(a) 1023
(b) 12