Question

determine the number of 5 cards combination out of 52 cards if each selection of 5 cards has exactly one king

Answer 1

Conditions

determine the number of 5 cards combination out of 52 cards if each selection of 5 cards has exactly one king

Solution

There are 4 kings in a deck of 52 cards. For each of 4 kings, we must calculate the number of combinations of 4 cards (5 cards minus one king) from a set of 48 cards (52 cards minus fixed king minus 3 other kings). The formula of such combination is below:

$$C _ {4 8} ^ {4} = \frac {4 8 !}{4 ! 4 4 !} = \frac {4 5 \cdot 4 6 \cdot 4 7 \cdot 4 8}{1 \cdot 2 \cdot 3 \cdot 4} = 1 9 4 5 8 0$$

As we have this number for all 4 kings, the answer is:

$$1 9 4 5 8 0 \cdot 4 = 7 7 8 3 2 0$$

Answer: 778320