Why is the probability of getting a pair in a five-card poker hand so complicated?

Question

The working-out of this question has really confused me. I know the basics of probability but I don't get the calculations here. I think 13C2 * 4C2 determines the number of possible pairs, and the stuff in the brackets is the number of combinations possible with the cards remaining after a pair is obtained—but, in them, what does does each number specifically do?

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Thanks in advance.

Answer 1

A simpler and equivalent way is the following:

The number of favourable events are:

$$13\times\binom{4}{2}\times\binom{12}{3}\times 4^3=1{,}098{,}240$$

this because:

• you have 13 different ways to choose the pair

• you have $\binom{4}{2}$ different suits for your chosen pair

• you have $\binom{12}{3}$ different ways to choose the 3 remaining cards

• you have $4^3$ different suits for the 3 remaining cards