Stuck on a Combination Problem

Question

We are trying to solve this combination problem:

In a box of 12 different colored crayons, one of them is black. In how many different ways can the teacher give these crayons to a student so that the student receives at least one black crayon? (Note: A student may receive 1 to 12 crayons.)

We tried getting the different possible combinations where you get at least 1 black crayon, such as 1 way only to get 1 black if you get 1 crayon, 11C1 ways to get at least 1 black if getting only 2 crayons, and so on. But I think we're going about it wrong. Would appreciate any help. Thanks in advance!

Answer 1

Hint: give the student the black crayon. Then you need to give the student some subset of the remaining crayons. How many is that?

Answer 2

black crayon only- $ 1$ way

black crayon & any other crayon -$11C1\,\,$ ways ( black is already chosen so only remaining 11 crayons are considered)

black crayon & any 2 other crayons -$11C2\,\,\,$ ways

follow this pattern.. finally add them all to get all possible ways.

Answer 3

The black crayons must be given. For other 11 crayons you can either give it or not. 2^11.

Correct me if i got the question wrong.