Cards

Given a standard, 52-card deck, how many combinations of five cards consist of at least three spades and exactly two cards of equal rank?

Originally posted by royalchicken
Given a standard, 52-card deck, how many combinations of five cards consist of at least three spades and exactly two cards of equal rank?

31086?

Maybe, maybe not. What is your reasoning?

Originally posted by royalchicken
Maybe, maybe not. What is your reasoning?

(1). # of 5-card combos wherein the pair are not one of the 3 spades = (# of distinct 3-spade combos) x (# of distinct non-spade pairs that dont match one of the spades + # of single spade pairs) = 66 x ((39 - 9) + (9x3)) = 3762

(2). # of combos wherein one of the pair is one of the 3 spades = (# of distinct 3-spade combos) x (# of possibilities for the 4th card for a given set of 3 spades) x (# of possibilities for the 5th card) = 66 x 9 x (48 - (2 (so that the 5th card doesn't match the pair) + 3 + 3 (so that the 5th card doesn't match either of the other two cards)) = 23760.

(1) + (2) = 27522.

(my original calculations didn't account for the possible matching of the pair with one of the spades)

I get 27522......Good job 😀! (My method was different though.)