How many different salads are possible?

Question

A salad bar has 7 ingredients, excluding the dressing. How many different salads are possible where 2 salads are different if they don't include identical ingredients?

Answer 1

7C1 ways to have only one ingredient

7C2 ways to have two ingredients

7C3 ways to have 3 ingredients, etc...

....

7C7 ways to have 7 ingredients.

 

The sum of all these can be represented in the following way, using the Binomial Theorem:

 

(1+1)⁷ - 7C0  (the latter wouldn't be a salad, lacking any ingredients) = 

2⁷ - 1 = 127.