But, that total is already less than the number of 5s of a kind. Now, that's a HUGE amount of overcounting I've just done - like the hand with A-2-3-4-5 of hearts plus the other two aces of hearts has just been counted 3 times. If we just naively count the number of choices for the values (A-5 through 10-A), which there are 10 of, times the 4 choices for the suits, times the 3 5 choices for which deck each one comes from, then we can choose 2 cards out of the remaining 151 we didn't use yet. But, we can overestimate the number as follows: The counting gets a little bit more complicated since we have to consider the possibility of having two copies of the same card from multiple decks, or of having a more than 5-card straight. Then we can count the possible straight flushes. The total of all three of these is 107,747,640. Then for 5 of a kind, we have to count the sets of 7 card hands that have 5, 6 or 7 cards of a single rank.ġ3 choices for the rank * 12C5 choices for which 5 we choose of those rank * 151C2 144C2 choices for the other 2 cards = 106,007,616 Let's assume the backs on the three decks are all different so we can tell them apart. Re-read your post before hitting submit, does it still make sense.Show your work! Detail what you have tried and what isn't working.Use proper spelling, grammar and punctuation.Give context and details to your question, not just the equation.Help others, help you! How to ask a good question Asking for solutions without any effort on your part, is not okay. Beginner questions and asking for help with homework is okay. Post your question and outline the steps you've taken to solve the problem on your own. Do not use ChatGPT in a question or an answerĭon't just post a question and say "HELP".Do not solicit or offer payments to complete your assignments or tests.No cheating - do not post questions from exams, tests, midterms, etc.No post flooding - Limit your posts to 2 or 3 questions a day.Don't be a jerk - don't be obnoxious or rude.Homework policy - asking for help is okay, asking to be given the solution is not. Make your question clear and concise - include steps you have tried.Stay on topic - this subreddit is for math questions no how-to guides, or non math related questions.
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