SOLUTION: A 3 card hand is drawn from a standard 52 card deck. a.) How many 3 card hands are possible? b.) How many hands, if all 3 cards must be of the same suit? c.) How many hand

Question 1127575: A 3 card hand is drawn from a standard 52 card deck.
a.) How many 3 card hands are possible?
b.) How many hands, if all 3 cards must be of the same suit?
c.) How many hands, if no 2 cards are of the same suit?
Answer by Edwin McCravy(20055) (Show Source): You can put this solution on YOUR website!
A 3 card hand is drawn from a standard 52 card deck.
a.) How many 3 card hands are possible?

That's 52 cards choose 3 = 52C3 = 22100 ways to choose 3 cards.

b.) How many hands, if all 3 cards must be of the same suit?

Choose the ranks: That's 13 ranks choose 3 = 13C3 = 286 ways
Choose the suit: That's 4 suits choose 1 = 4C1 = 4 ways
Answer: 286∙4 = 1144 ways

c.) How many hands, if no 2 cards are of the same suit?

Think of the 4 suits in alphabetical order:
Clubs, Diamonds, Hearts, Spades.

Choose the 3 suits:
That's 4 suits choose 3 = 4C3 = 4 ways.

Choose the rank for the card with the suit that starts with the letter
closest to the first of the alphabet.

That's 13 ranks choose 1 = 13C1 = 13 ways

Choose the rank for the card with a suit that starts with the letter
next to the closest to the first of the alphabet.

That's 12 remaining ranks choose 1 = 12C1 = 12 ways.

Choose the rank for the card with a suit that starts with the letter
closest to the end of the alphabet.

That's 11 remaining ranks choose 1 = 11C1 = 11 ways.
 
That's 4∙13∙12∙11 = 6864

Edwin

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