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I am CS student and I have a Qualifying exam this September and I hope someone help me in this question:

Suppose a 13-card bridge hand is drawn from a deck of 52 cards. How many ways are there to obtain :

a. 13 cards of the same suit?

b. 5 spades, 4 clubs, 3 hearts, 1 diamond?

c. 5 cards of one suit, 4 cards of a second suit, 3 cards of a third suit and 1 card of a fourth suit?

d. exactly 3 aces?

e. 13 cards no two of which have the same number?:

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What have you tried? –  joriki Jul 29 '12 at 21:55
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First you have to decide whether order matters: it will make a difference of a factor of $13!=6227020800$ to your answers –  Henry Jul 29 '12 at 22:36
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@Henry, it's a bridge hand, so presumably all that matters is the set of cards you wind up with, not the order in whcih you acquire them. –  Gerry Myerson Jul 30 '12 at 0:02
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1 Answer

HINTS

I'll do a few, and we'll see if they strike a chord within you.

  1. For (a), there are exactly 13 cards in a suit. There are 4 suits. So you will draw all the cards from 1 of the 4 suits. So we have as many options as there are suits.
  2. For (e), there are 13 different 'numbered' cards, where I assume that two jacks have the same 'number.' For each 'numbered' card, there are 4 choices. Thus I choose between 4 things a total of 13 times.
  3. For (d), I might decompose this. Suppose I am dealt 3 cards that are aces, and then 10 other non-ace cards. There are 48 non-ace cards in the deck, and 4 aces. So how many ways can I get 3 aces from a pool of 4 aces, and how many ways can I get 10 other cards from a pool of 48 cards?
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