How many 5-card hands from a standard 52-card deck contain exactly 3 jacks and exactly 2 hearts?
The solution I came up with is C(13,3)*C(13,2)*C(5,2) but I think I'm wrong. What would be the smart way to approach this problem and find the answer?
[Math] How many 5-card hands from a standard 52-card deck contain exactly 3 jacks and exactly 2 hearts
combinatoricsprobability
Best Answer
all the ways you can have 3 jacks that arent a heart, and 2 hearts: $\binom{3}{3}$ (ways to choose three jacks from three non heart jacks) $\times \binom{12}{2}$ (ways too choose two hearts from the twelve hearts that arent a jack). Plus all the ways you can have 3 jacks one of which is a heart and two hearts: $\binom{1}{1}$ (ways to choose the jack of hearts) $\times \binom{3}{2}$ (ways to choose two jacks from three non heart jacks) $\times \binom{12}{1}$ (ways to choose the remaining heart from the twelve non heart jacks) $\times \binom{36}{1}$ (ways to choose the final card from the thirty-sex cards that arent a jack or a heart).
so $\binom{3}{3}\binom{12}{2} + \binom{1}{1}\binom{3}{2}\binom{12}{1}\binom{36}{1}$