Understanding Sample Spaces & Card Shuffling

[EugP]

Homework Statement

I'm having some trouble understanding how to write a sample space in a problem.
Here's an example:
Shuffle a deck of cards and turn over the first card. What is the sample space of this experiment? How many outcomes are in the event that the first card is a heart?

Homework Equations

$$C_k^n = {n \choose k} = \frac{n!}{k!(n - k)!}$$

The Attempt at a Solution

From what I was explained, sample space is the mutually exclusive and collectively exhaustive set of all possible outcomes. So in my case, wouldn't it be {2-A of hearts, 2-A of spades, 2-A of clubs, 2-A of diamonds} ? Those together create all the possiblities in the deck.
For the second part, isn't it simply 52 choose 13? If it is, it will just be

$${52 \choose 13} = \frac{52!}{13!(52 - 13)!} = 635,013,559,600 \approx 11 6.350135596 \cdot 10^{11}$$

But I'm not sure this is right. There are no answers in my book. If someone could help me on this it would be greatly appreciated.

 

[matness]

yes. You are right.
Sample space consists of all cards in the deck
and the answer of the second part is in mattmns 's post

 

[EugP]

Thanks for verifying!

 

[mattmns]

I disagree with your second answer, I think it should be 13. The possible outcomes of a heart being the first card flipper over are A hearts, K hearts, ... , 2 hearts.