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?
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?