Determine the number of 5-card combinations

Question:

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:

From a deck of 52 cards, 5-card combinations have to be made in such a way that in each selection of 5 cards, there is exactly one king.

In a deck of 52 cards, there are 4 kings.

1 king can be selected out of 4 kings in ${ }^{4} \mathrm{C}_{1}$ ways.

4 cards out of the remaining 48 cards can be selected in ${ }^{48} \mathrm{C}_{4}$ ways.

Thus, the required number of 5-card combinations is ${ }^{4} \mathrm{C}_{1} \times{ }^{48} \mathrm{C}_{4}$.

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