# Homework Help: Counting question - check reasoning

1. Jul 7, 2006

### Magotine

Hi All,

I'm taking discrete math as part of my computer science course. i don't quite understand why i didn't get the answer in the book.

Please take a look at my attachment and see where have i gone wrong.

Thanks.

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2. Jul 8, 2006

### HallsofIvy

How many ways are there of picking
a) A King and a red card
b) A King or a red card
from a standard deck of 52 cards?
I find the question ambiguous. Are you drawing a single card and asking "how many ways are there of doing this so that the card is both red and a King (b: either red or a King)?" or are you drawing two cards and asking "how many ways are you doing this so that one of the cards is a King and the other red (b: one of the cards is a King or one of the cards is red)?"

If it were the first, then there are exactly two "red Kings" so the answer would be 2. Since you say the correct answer is 102, I guess that's not what is being asked.

To answer the question you asked, no, I do not believe the wording requires that the King not be a red card however, since it says
"a King and a red card", drawing a red King and a black card would not qualify as such.

Okay, so you draw two cards. There are 4 ways the first can be a King. But now we have to be careful. If the first card is a red King, then the ways a red card can be drawn next are reduced.
There are 2 ways the first card can be a black King. Then there are 26 ways the second card can be red: 2(26)= 52. There are 2 ways the first card can be a red King. Then there are 25 ways the second card can be red: 2(25)= 100. I'm assuming that drawing a red card first and a King second is not considere a "different way" from drawing the same cards, King first.

Oddly enough, the only way I can get the answer you give for b), 28, is to reverse the assumption I made in a)! Obviously, the number of ways we can do "A or B" is greater than the number of ways we can do "A and B" as long as we have the same situation! If we are drawing two cards then "getting a King or a red card" would have to mean that "at least one of the cards is red or at least one of the cards is a King" and there are many more ways of doing that than 28.

Assuming b) is referring to drawing a single card, how many ways can we draw so that that card is either red or a King? Well, how many red cards are there? How many non-red Kings are there?