You are dealt cards from a standard deck of 52 cards. How many ways can you be dealt the cards so that they contain two cards of one rank, two cards of another rank, and a fifth card of a third rank? We say that such a hand has two pairs. For example, the hand QQ225 has two pairs. (Assume that the order of the cards does not matter.)

I went over this a bazillion times and it's STILL WRONG. Plz help :(

SavvyBee May 15, 2020

#1**0 **

There are 13*3 possibilities for the first pair, leaving 12*3 possibilities for the 2nd pair, leaving 11*4 possibilities for the singleton. So: 13*3*12*3*11*4 = 61776 in total.

Guest May 15, 2020

#2**+2 **

SavvyBee, I recall this is an AoPS intro to counting quesiton, please provide problem number and week so I can give you a hint.

hugomimihu May 15, 2020

#3**0 **

Yeah, I've gotten this wrong a ton of times. Thank you! It's Week 3 right now, problem 7.

SavvyBee May 15, 2020

#4**+1 **

Sorry for the late reply.

We choose the two paired ranks in \(\binom{13}{2} = 78\) ways and the remaining rank in \(\binom{11}{1} = 11\) ways.

Now you know this calculate the number of ways you can pick the suit.

Good luck!

Your effort to get that 1 point is commendable though.

Me: That's just 1 point. Meh.

hugomimihu
May 15, 2020