If four cards are chosen from a pack of 52 playing cards then find the number of ways that all four cards are:
a) of same suit
b) red
c) face cards
d) king
e) of different suit
Solution:
a) All four cards are of the same suit:
There are indeed 4 suits, and for each suit, there are 13 cards. So, the number of ways to choose 4 cards of the same suit should be:
\[4C1 \times 13C4 = 4 \times \frac{13!}{4!(13-4)!} = 4 \times \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 4 \times 715 = 2860\]
So, the answer is indeed 2860 ways.
b) All four cards are red:
There are 26 red cards in total (13 hearts and 13 diamonds). So, the number of ways to choose 4 red cards should be:
\[26C4 = \frac{26!}{4!(26-4)!} = \frac{26 \times 25 \times 24 \times 23}{4 \times 3 \times 2 \times 1} = 14,950\]
So, the answer is indeed 14,950 ways.
c) All four cards are face cards:
There are 3 face cards in each suit, making a total of \(3 \times 4 = 12\) face cards in the deck. So, the number of ways to choose 4 face cards should be:
\[12C4 = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495\]
So, the answer is indeed 495 ways.
d) All four cards are kings:
There are indeed 4 kings in total. So, the number of ways to choose all 4 kings should be:
\[4C4 = 1\]
So, the answer is indeed 1 way.
e) All four cards are of different suits:
There are 4 suits, and for each suit, there are 13 cards. So, the number of ways to choose one card from each suit should be:
\[13^4 = 28,561\]
So, the answer is indeed 28,561 ways.
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