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October 17th, 2018, 09:28 AM  #1 
Newbie Joined: Oct 2018 From: Croatia Posts: 4 Thanks: 0  Permutations problem
This is a problem that has come up in my math test recently and I'm unsure whether the teacher's solution is correct: There are 12 people and 3 cars. Each car can fit 4 people. If the owner of the car drives their car, in how many ways could the rest 9 people be distributed in the cars? This would mean that there is one set person per car, so there are 3 spots in every car to fill. The order in which they are sat in the car doesn't matter. The answer I got is: ((9 above 3)*(6 above 3))/3! The answer my teacher has is: ((9 above 3)*(6 above 3))*3! Please leave a detailed response as to which answer is correct because, if mine is correct, I will have to dispute it with her so I need all the details I can get. Thanks in advance! Last edited by skipjack; October 19th, 2018 at 10:02 AM. 
October 17th, 2018, 01:51 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,762 Thanks: 697 
What does 9 above 3 mean? Is it $\frac{9!}{6!3!}$?

October 17th, 2018, 02:37 PM  #3 
Newbie Joined: Oct 2018 From: Croatia Posts: 4 Thanks: 0 
Yes, by 9 above 3 i mean 9! divided by 6!*3! but I am not familiar with the formatting on this website.

October 17th, 2018, 03:20 PM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,417 Thanks: 1025  To make sure: what is actual results of those 2 calculations?
Last edited by Denis; October 17th, 2018 at 04:14 PM. 
October 17th, 2018, 09:20 PM  #5 
Newbie Joined: Oct 2018 From: Croatia Posts: 4 Thanks: 0 
My result is 280 ways, and my teachers is 10080.

October 18th, 2018, 03:55 AM  #6 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 140 
I think the answer depends on whether the cars are distinguished. I'll assume that the cars are somehow distinguished  1st, 2nd, 3rd or red, green, white, etc. I get $\displaystyle \binom{9}{3} \cdot \binom{6}{3} \cdot 3!=10080$ which agrees with your teacher. Call the 9 nondrivers A,B,C,D,E,F,G,H,J. The reason for multiplying by 3! at the end is because we want the arrangement {A,B,C} {D,E,F} {G,H,J} to be different from {D,E,F} {A,B,C} {G,H,J} since the cars are distinguishable and there are 3! ways of ordering the cars. 
October 18th, 2018, 03:59 AM  #7 
Newbie Joined: Oct 2018 From: Croatia Posts: 4 Thanks: 0 
Yeah I see, thank you for clearing that up. She isnâ€™t really good at explaining the permutations, and I remembered a similar task we had in ozr workbook but alas, they were not similar enough.

October 18th, 2018, 05:39 AM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,417 Thanks: 1025  
October 18th, 2018, 08:19 AM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,417 Thanks: 1025 
Changing your problem's wording: 3 boxes are labelled 1,2,3. 9 cards are labelled 1 to 9. 3 cards are put in each box; the order within a box does not matter. In how many ways can this be done? Answer is 1680: [Ways][Box1][Box2][Box3] [0001][..123][..456][..789] [0002][..123][..457][..689] [0003][..123][..458][..679] [0004][..123][..459][..678] ..... [1677][..789][..345][..126] [1678][..789][..346][..125] [1679][..789][..356][..124] [1680][..789][..456][..123] So 1680 ways if order does not matter within box, but the boxes are distinguishable. 1680 / 6 = 280 : your answer 1680 * 6 = 10080 : teacher's answer So you guys get together and decide how to fix the terrible wording of the problem! 
October 18th, 2018, 01:22 PM  #10 
Global Moderator Joined: May 2007 Posts: 6,762 Thanks: 697 
I believe both of you are wrong. The correct answer is $\binom{9}{3}\times \binom{6}{3}=\frac{9!}{(3!)^3}$. You teacher's derivation is almost correct, but the final multiplication by$3!$ is wrong. Rearranging the order the cars doesn't add any more possibilities. Example: cars are a,b,c and people 19. Let us use cars in order a,b,c initially and b,c,a alternatively. Consider a sort with (1,2,3) in b, (4,5,6) in c, and (7,8,9) in a. However in the initial sort we have the possibility of (7,8,9) in a, (1,2,3) in b, and (4,5,6) in c. This shows that the initial car order covers all possibilities. An alternative way of getting the result is considering all possible permutations of 9 people $9!$ and place them 3 at a time in the cars. Since the order within a specific car does not matter, you need to divide by $3!$ for each car, ending up with $\frac{9!}{(3!)^3}$. 

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combination, factorial, permutations, problem 
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