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: $c(17,0) + c(17,1) + c(17,2) + ... + c(17,16) + c(17,17)$

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

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Woops- I made a mistake there. Supposed to be multiply 1) and 2) – Arvin May 22 '11 at 4:57

As for fewer calculations: what you need to do is pick a subset of the remaining 17 shows to represent the shows you have heard of. There are $2^{17}$ possible subsets, so that's what you want. Alternatively, for each of the remaining 17 programs, you can either have heard of it before or not; so you have one of two choices for each of the remaining 17 programs. That means making a choice from 2 possibilities, 17 times, or $2^{17}$ possibilities.
And alternatively, $$C(17,0) + C(17,1) + \cdots + C(17,17) = (1+1)^{17} = 2^{17}$$ by the binomial theorem, so that's another way to see that the big sum you have is simply $2^{17}$.
So the correct answer is $2^{17}\times\binom{20}{3}$.