Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
Solution and Explanation
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 \(^4C_1\) ways and the remaining 4 cards can be selected out of the 48 cards in \(^{48}C_4\) ways.
\(=\space^{48}C_4\times\space^4C_1=\frac{48!}{4!44!}\times\frac{4!}{1!3!}\)
\(=\frac{48\times47\times46\times45}{4\times3\times2\times1\times4}\)
Thus, by multiplication principle, required number of 5 card combinations=778320
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Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
- Combination refers to the combination of about n things taken k at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.