Just want to check if my answer and reasoning is correct for the following problem (Not a homework problem - it is a sample question for a test I'm preparing for)

In a survey, viewers were given a list of 20 TV Shows and are asked to label 3 favourites not in any order. Then they must tick the ones that they have heard of before, if any. How many ways can the form be filled, assuming everyone has 3 favourites?

My reasoning:

1) Choose 3 shows out of 20: c(20,3)

2) Choosing 0-17 shows from 17 choices: $c(17,0)+c(17,1)+c(17,2)+...+c(17,16)+c(17,17)$

and add 1) and 2) together for the final answer.

Would this be correct? Is there a better way of doing the second part that doesn't involve so many calculations?

In a survey, viewers were given a list of 20 TV Shows and are asked to label 3 favourites not in any order. Then they must tick the ones that they have heard of before, if any. How many ways can the form be filled, assuming everyone has 3 favourites?

My reasoning:

1) Choose 3 shows out of 20: c(20,3)

2) Choosing 0-17 shows from 17 choices: $c(17,0)+c(17,1)+c(17,2)+...+c(17,16)+c(17,17)$

and add 1) and 2) together for the final answer.

Would this be correct? Is there a better way of doing the second part that doesn't involve so many calculations?