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February 1st, 2018, 11:30 PM  #1 
Member Joined: Feb 2018 From: Canada Posts: 42 Thanks: 2 
What is the probability of being dealt a bridge hand with exactly 4 honour cards and exactly 4 cards from the 5 through 10? Each hand is dealt 13 cards. Honour cards: 20 cards (A, K, Q, J, 10) From 5 through 10: 24 cards (10, 9, 8, 7, 6, 5) Others: 12 cards (4, 3, 2) I am thinking about 5 cases. Case1: No 10: (16C6*20C4*12C5)/52C13. Case2: One 10: (4C1*16C3*20C3*12C6)/52C13. Case3: Two 10's: (4C2*16C2*20C2*12C6)/52C13. Case4: Three 10's: Case5: Four 10's: Then add them up. But I have a feeling that I am doing it wrong. Please help. Thank you. Last edited by skipjack; February 2nd, 2018 at 04:48 AM. 
February 2nd, 2018, 12:23 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198 
this is a multinomial 3 classes: Honor cards: 16 (I'm assuming 10 isn't an honor card as you labeled it) 5  10: 24 Others: 12 $\dfrac{\dbinom{16}{4}\dbinom{24}{4}\dbinom{12}{5} }{\dbinom{52}{13}} \approx 0.024$ Last edited by romsek; February 2nd, 2018 at 12:26 AM. 
February 2nd, 2018, 06:33 AM  #3 
Member Joined: Feb 2018 From: Canada Posts: 42 Thanks: 2 
Thanks for your help but 10 is one of the honour cards that's why this question has multiple cases. I know it has multiple cases but but I don't know how to solve. 10 is overlapped between honour group cards and from the 5 to 10 group cards. Please help. 
February 2nd, 2018, 09:24 AM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198 
ok let's use 4 classes then honor cards: 16 JA 10's: 4 59: 20 others:12 Now apply your cases (all examples are divided by 52C13) 0 10s : (16C4)(4C0)(20C4)(12C5) 1 10 : (16C3)(4C1)(20C3)(12C6) 2 10s : (16C2)(4C2)(20C2)(12C7) 3 10s : (16C1)(4C3)(20C1)(12C8 ) 4 10s : (16C0)(4C4)(20C0)(12C9) and add all these you were quite close, ring back if you don't understand all the numbers 
February 2nd, 2018, 02:26 PM  #5 
Member Joined: Feb 2018 From: Canada Posts: 42 Thanks: 2 
I have just gotten exactly what you did. Thank you very much for your help. I really appreciate for your help. I am quite new here. Where can I give you a thank. 
February 2nd, 2018, 02:27 PM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198  

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