Based on the rule of symetry, the Carmelo and LeBron will defeat eachother with SAME number.

So 7!/2 = 2520

[quote="logitech"]Based on the rule of symetry, the Carmelo and LeBron will defeat eachother with SAME number.

So 7!/2 = 2520[/quote]

I don't get your reasoning... the question only asks only the different arrangement in which Carmelo defeats LeBron. I think you've added the possibility that Lebron defeats carmelo too.

Can you further explain your reasoning?

thanks

First, the total number of arrangements is 7!

Among the arrangments, the number of arrangements in which Carmelo defeats LeBron and Carmelo defeats Lebron must be the same as there are no ties.

So the number of arrangements of finishes in which Carmelo defeats LeBron must be 7!/2; similarly, the number of arrangements of finishes in which LeBron defeats Carmelo must also be 7!/2.

I hope the explanation is clear.

C - carmelo
L - lebron

Case 1: carmelo is 1st
C X X X X X X
Lebron can take any of the 6 places

Case 1: carmelo is 2nd
Y C X X X X X
Lebron can take any of the 5 places

Case 1: carmelo is 3rd
Y Y C X X X X
Lebron can take any of the 4 places

Case 1: carmelo is 4th
Y Y Y C X X X
Lebron can take any of the 3 places

Case 1: carmelo is 5th
Y Y Y Y C X X
Lebron can take any of the 2 places

Case 1: carmelo is 6th
Y Y Y Y Y C X
Lebron can take any of the 1 place

The answer is: 6 x 5 x 4 x 3 x x 1 = 720

fair enough?

tini, logitech showed the best and shortest way for this problem.

Your algorythm has some mistakes.
According to your statement if C is 1st, L can take any of the 6 places. But here is more than 6 combinations, because other team members can change their places too.
So in each case you should at first count the ways you can allocate L and then multiply it on the ways you can allocate other 5 contestants.
After that you should sum up all 6 cases.

The answer is B. 2520.

2520 = 6x5!+5x5!+4x5!+3x5!+2x5!+1x5!

logitech wrote:Based on the rule of symetry, the Carmelo and LeBron will defeat eachother with SAME number.

So 7!/2 = 2520

A silly question, but would this be also true if the number of participants were in even number? How would someone proceed then?

kanha81 wrote:

logitech wrote:Based on the rule of symetry, the Carmelo and LeBron will defeat eachother with SAME number.

So 7!/2 = 2520

A silly question, but would this be also true if the number of participants were in even number? How would someone proceed then?

Yes.

"The rule of symmetry" is just a fancy way of saying that there will be an equal number of arrangements in which C beats L as in which L beats C. So, if we only want the arrangements in which C beats L, the answer will always be:

(Total # of arrangements)/2

Stuart Kovinsky wrote:

kanha81 wrote:

logitech wrote:Based on the rule of symetry, the Carmelo and LeBron will defeat eachother with SAME number.

So 7!/2 = 2520

A silly question, but would this be also true if the number of participants were in even number? How would someone proceed then?

Yes.

"The rule of symmetry" is just a fancy way of saying that there will be an equal number of arrangements in which C beats L as in which L beats C. So, if we only want the arrangements in which C beats L, the answer will always be:

(Total # of arrangements)/2

Thanks a bunch Stuart!

ikhabibullin wrote:tini, logitech showed the best and shortest way for this problem.

Your algorythm has some mistakes.
According to your statement if C is 1st, L can take any of the 6 places. But here is more than 6 combinations, because other team members can change their places too.
So in each case you should at first count the ways you can allocate L and then multiply it on the ways you can allocate other 5 contestants.
After that you should sum up all 6 cases.

The answer is B. 2520.

2520 = 6x5!+5x5!+4x5!+3x5!+2x5!+1x5!

Thats a cool explanation. Thank you

billzhao wrote:First, the total number of arrangements is 7!

Among the arrangments, the number of arrangements in which Carmelo defeats LeBron and Carmelo defeats Lebron must be the same as there are no ties.

So the number of arrangements of finishes in which Carmelo defeats LeBron must be 7!/2; similarly, the number of arrangements of finishes in which LeBron defeats Carmelo must also be 7!/2.

I hope the explanation is clear.

This is not the best explanation you provided for this solution. 7!/2! Can be explained this way: If the seven slots are arranged thus:

1 2 3 4 5 6 7 , where 1 means the first place and 2 means 2nd and so on, then it is clear from the problem that Lebron can never be in 1. Now for all (C, L) pairs, L can never come before C. Now in permutation whenever there are restrictions involving pairs of things, we use 2! to isolated these cases. So this explains the short method above, since there are 7! ways of arranging all of them.

For a simpler example, say that A B C are to seated in a row but A can never come before B.

A B C
B A C
C B A
C A B
B C A
A C B

If we implemented this restriction we will be left with,

B A C
C B A
B C A

We got here by 3!/2! = 3. All cases involving A coming before B (Counting from the left) have been eliminated by diving by 2!.