Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

Solution

Total cards = 52
Number of cards to be selected = 5
Type of cards requried = 1 king + 4 others
We have 4 kings + 48 others.
$rightwards double arrow$ Number of selections = $straight C presuperscript 4 subscript 1 cross times straight C presuperscript 48 subscript 4 space equals space fraction numerator 4 space factorial over denominator 3 factorial space 1 factorial end fraction cross times fraction numerator 48 space factorial over denominator 44 factorial space 4 factorial end fraction$
= $4 cross times fraction numerator 48 cross times 47 cross times 46 cross times 45 over denominator 4 cross times 3 cross times 2 cross times 1 end fraction equals 4 cross times 2 cross times 47 cross times 46 cross times 45$
= 778320

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Permutations And Combinations

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

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