Combinatorics: Ways to Choose Playing Cards

[Shoney45]

Homework Statement

How many ways are there to pick 2 different cards from a standard 52-card deck such that the first card is a spade and the second card is not a queen.

Homework Equations

P(n,k) C(n,k)

The Attempt at a Solution

Since the player must have one spade, there are 13 ways to deal a spade. I now have 51 cards to deal out.

I am mixed up on what to do with the second card though. I can't find a way to deal with the case where the first spade that I deal out happens to be a queen. If it was a queen, then I still need to take out three other queens from the deck, leaving me with 48 cards. If the first card wasn't a queen, then I still need to take out four cards from the deck, leaving me with 47 cards. 13 x 48 is a different result than 13 x 47. But I know that I am only supposed to have one answer. So I really don't know how to proceed on this one.

 

[Robert1986]

Homework Statement

How many ways are there to pick 2 different cards from a standard 52-card deck such that the first card is a spade and the second card is not a queen.

Homework Equations

P(n,k) C(n,k)

The Attempt at a Solution

Since the player must have one spade, there are 13 ways to deal a spade. I now have 51 cards to deal out.

I am mixed up on what to do with the second card though. I can't find a way to deal with the case where the first spade that I deal out happens to be a queen. If it was a queen, then I still need to take out three other queens from the deck, leaving me with 48 cards. If the first card wasn't a queen, then I still need to take out four cards from the deck, leaving me with 47 cards. 13 x 48 is a different result than 13 x 47. But I know that I am only supposed to have one answer. So I really don't know how to proceed on this one.

Note that the first card is either a Queen of Spades or it isn't. Thus, to count the total number of ways to do this, you can count how many ways there are to pick cards such that the first card is a non-queen spade and the second is a non-queen, then count the number of ways there are to pick the queen of spade and then a non-queen card. Then add those two number together.

 

[Shoney45]

Note that the first card is either a Queen of Spades or it isn't. Thus, to count the total number of ways to do this, you can count how many ways there are to pick cards such that the first card is a non-queen spade and the second is a non-queen, then count the number of ways there are to pick the queen of spade and then a non-queen card. Then add those two number together.

Okay, I think I got it then. So if I give the player a spade that does not happen to be a queen, and then I take out the four queens, I am left with (12 x 47). If he happens to receive the queen of spades first, and I take out the remaining three queens, that equals (1 x 48). So the answer is (12 x 47) + (1 x 48). Does that look about right?

 

[Robert1986]

Okay, I think I got it then. So if I give the player a spade that does not happen to be a queen, and then I take out the four queens, I am left with (12 x 47). If he happens to receive the queen of spades first, and I take out the remaining three queens, that equals (1 x 48). So the answer is (12 x 47) + (1 x 48). Does that look about right?

Yep, looks right to me.