Eight cards are chosen at random from a standard deck of playing cards. In how many ways can the cards be chosen if three or more cards are kings.
A)6,849,216
B)7,043,796
C)73,629,072
D)194,580
E)294,516,288

Question

Quizplus · Accepted Answer

To solve this problem, we consider the cases where there are exactly 3, 4, or all 4 kings among the chosen cards, since a standard deck has 4 kings.1. **Exactly 3 Kings**: Choose 3 kings from the 4 available, and the remaining 5 cards from the 48 non-kings. This gives us combinations of $ \binom{4}{3} \times \binom{48}{5} $.2. **Exactly 4 Kings**: Choose all 4 kings, and the remaining 4 cards from the 48 non-kings. This gives us combinations of $ \binom{4}{4} \times \binom{48}{4} $.Adding these two cases together gives the total number of ways to choose the cards:$$ \binom{4}{3} \times \binom{48}{5} + \binom{4}{4} \times \binom{48}{4} $$$$ = 4 \times 1,712,304 + 1 \times 194,580 $$$$ = 6,849,216 + 194,580 $$$$ = 7,043,796 $$Therefore, the correct answer is 7,043,796.

Eight Cards Are Chosen at Random from a Standard Deck