SOLUTION: A game uses the 10, jack, queen, king, and ace of hearts and spades. How many 6-card hands contain a) an equal number of hearts and spades? b) more hearts than spades?

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Question 1191644: A game uses the 10, jack, queen, king, and ace of hearts and spades. How many 6-card hands contain
a) an equal number of hearts and spades?
b) more hearts than spades?

Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The total number of ways of choosing 6 of the 10 cards is

C%2810%2C6%29=210

The number of ways of choosing 3 of the 5 cards in each suit is

C%285%2C3%29%2AC%285%2C3%29=10%2A10=100

ANSWER part a: 100

By symmetry, the number of ways of getting more hearts than spades is equal to the number of ways of getting more spades than hearts. So the number of ways of getting more hearts than spades is

%281%2F2%29%28210-100%29+=+55.

ANSWER part b: 55

Note we can find the answer to part b with more straightforward methods.

4 hearts and 2 spades: C%285%2C4%29%2AC%285%2C2%29=5%2A10=50
5 hearts and 1 spade: C%285%2C5%29%2AC%285%2C1%29+=+1%2A5=5

More hearts than spades: 50+5 = 55