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Question
Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
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Solution
In a deck of 52 cards, there are 4 aces. A combination of 5 cards have to be made in which there is exactly one ace.
Then, one ace can be selected in 4C1 ways and the remaining 4 cards can be selected out of the 48 cards in 48C4 ways.
Thus, by multiplication principle, required number of 5 card combinations
= `""^48C_4 xx ""^4C_1 = (48!)/(4! xx 44!) xx (4!)/(1! xx 3!)`
= `(48 xx 47 xx 46 xx 45)/(4 xx 3 xx 2 xx 1) xx 4`
= 778320
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