8 8 Cars in 3 3 Trailers

A trailer can be loaded with 1 car, or with 4 cars.

780 770 750 760

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3 solutions

Happy Melodies
Mar 3, 2014

Motivation When we see such combinatorics question with a maximum number of items being placed into a box, we usually think of using the maximum, or m a x max idea as shown below:

Note The 3 3 trailers are identical.

If the m a x max of the number of cars in the trailers is 3 3 , we will have only 1 1 case to consider:

  • ( 3 , 3 , 2 ) (3,3,2) - referring to 3 3 cars in 2 2 trailers and 2 2 cars in the remaining trailer.

The number of ways to arrange the 8 8 cars in this order is simply: ( 8 2 ) ( 6 3 ) 2 = 280 \dbinom{8}{2} \cdot \dfrac{\binom{6}{3}}{2}= 280 . We choose 2 2 cars out of 8 8 to put into 1 1 trailer, then we simply choose 3 3 out of the remaining 6 6 cars, then divide by 2 2 because choosing 3 3 cars c 1 , c 2 , c 3 c_1,c_2,c_3 to put into one trailer is the same as choosing c 1 , c 2 , c 3 c_1,c_2,c_3 to put into the other trailer.

If the m a x max is 4 4 , there are 2 2 cases to consider:

  • ( 4 , 3 , 1 ) (4,3,1) : Number of ways to arrange = ( 8 4 ) ( 4 3 ) = 280 = \dbinom{8}{4} \cdot \dbinom{4}{3} = 280

  • ( 4 , 2 , 2 ) (4,2,2) : Number of ways to arrange ( 8 4 ) ( 4 2 ) 2 = 210 \dbinom{8}{4} \cdot \dfrac{\binom{4}{2}}{2}= 210 .

Therefore, the total number of ways is 280 + 280 + 210 = 770 280 +280+210 = \boxed{770} . We are done. :)

The problem said between 1 and 4, so it could only have 2 or 3, so shouldn't the answer also be 280?

Albert Xu - 7 years, 3 months ago

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Sorry but what do you mean by 2 2 or 3 3 and the answer also 280 280 ?

Happy Melodies - 7 years, 3 months ago

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Since the number of cars per trailer has to be between 1 and 4, only 2 cars or 3 cars per trailer are allowed. There are 280 ways to do this.

Albert Xu - 7 years, 3 months ago

It is not "strictly between 1 and 4", so you are allowed to have 1 car or 4 cars.

I'd add in an explanation for clarity.

Calvin Lin Staff - 7 years, 3 months ago

yeah the answer is 280

Darren Huang - 7 years, 3 months ago

banglaralobd.com

Masud Milon - 7 years, 3 months ago

this question of which standerd

Haji Khalil - 7 years, 3 months ago

same as my approach except for the division by 2 to take care of the symmetry (which was my mistake). In the first case, i suppose we could have calculated the same as 8C3 * 6C2 / 2.. Also, aren't there 3 ways in which the 3 , 3 and 2 can be arranged (3, 3 and 2 / 2, 3 and 3 and 3, 2 and 3) thus requiring division by 3

Sundar R - 7 years, 3 months ago

what's that operator (8 2) that u have used in first step to get 280??

Ak Sharma - 7 years, 2 months ago
Dennis Gulko
Mar 3, 2014

This is equivalent to the number of 8 8 letter words consisting of A , B , C A,B,C where each letter appears between 1 1 and 4 4 times - by writing down to which trailer each car goes, and then dividing by 3 ! 3! , since the trailers are identical.

We will do it using exponential generating functions: for each letter, f X ( x ) = x + x 2 2 ! + x 3 3 ! + x 4 4 ! f_X(x)=x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} - for X = A , B , C X=A,B,C . So the generating function is f ( x ) = f A ( x ) f B ( x ) f C ( x ) = ( x + x 2 2 ! + x 3 3 ! + x 4 4 ! ) 3 f(x)=f_A(x)\cdot f_B(x)\cdot f_C(x)=\left(x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}\right)^3 And we are looking for the coefficient of x 8 8 ! \frac{x^8}{8!} , which is 4620 4620 - by direct computation while remembering that we only need x 8 x^8 .

So, the number of 8 8 letter words as described above is 4620 4620 , and so the answer will be 4620 / 3 ! = 770 4620/3!=\boxed{770}

Can you please explain the concept of exponential g . f. ?

A Brilliant Member - 7 years, 3 months ago
Dale Steyn
Feb 21, 2014

1) ( 8 4 ) ( 4 3 ) ( 1 1 ) = 280 {8 \choose 4}*{4\choose 3}*{1 \choose 1} = 280 (4 3 1)
2) ( 8 4 ) ( 4 2 ) ( 2 2 ) = 420 {8 \choose 4}*{4\choose 2}*{2 \choose 2} = 420 (4 2 2)
3) ( 8 3 ) ( 5 3 ) ( 2 2 ) = 560 {8 \choose 3}*{5 \choose 3}*{2\choose 2} = 560 (3 3 2)

Total number of ways = ( 1 ) + ( 2 ) + ( 3 ) 2 = 770 = (1) + \frac{ (2) +(3)} {2} = \boxed{770} .

Can you explain why "Total = ( 1 ) + ( 2 ) + ( 3 ) 2 = (1) + \frac{ (2) + (3) } {2} ? Many people feel that it should be ( 1 ) + ( 2 ) + ( 3 ) (1) + (2) + (3) for a total of 1260.

Note: To type latex, you just need to add the brackets \ ( and \ ) with spaced removed. You can see how I edited your solution, as an example.

Calvin Lin Staff - 7 years, 3 months ago

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In case (2) you are counting the train with 2 cars twice and in case (3) you are counting twice the train with 3 cars. Because of that, you have to divide by 2.

Igor dos Santos Silva - 7 years, 3 months ago

i require some more clear explanation can some help in getting out of this jugulary????

Rahul Sharma - 7 years, 3 months ago

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