HELP NEEDED !!! DISCRETE MATHS

Question

In the game of poker, a player receives a subset of 5 cards, called a poker hand, from the standard deck of 52 cards. The order in which the cards are received is not important, just the actual cards themselves.



How many poker hands contain exactly two suits?

Answer 1

2(₄C₂)(₁₃C₁)(₁₃C₄₎ + 2(₄C₂)(₁₃C₂)(₁₃C₃) = 2*6*13*715 + 2*6*78*286= 379236. You can have exactly two suits in a poker hand by choosing 1 card from one of the suits and 4 cards from another suit or choosing 2 cards of from one of the suits and 3 cards from another suit. Therefore, you have two terms involving (₁₃C₁)(₁₃C₄) and (₁₃C₂)(₁₃C₃). You multiply them by 2(₄C₂) because you need to pick cards from two of the four suits and include the two possible cases where you have one suit being chosen more often over the other. Hopefully, this helps.