World Cup shootout

[jay_shark]

The following procedure is used to break ties in a world cup soccer tournament . Each team selects 5 players in a prescribed order to take a penalty kick . Each of these players takes a penalty kick, with a player from the first team followed by a player from the second team and so on .

How many different scoring scenarios are possible if the game is settled in the first round of 10 kicks ? The round ends once it is impossible for a team to equal the number of goals scored by the other team .

[Enrique]

You can have 25 different scoring scenarios:
0-0, 1-0, 0-1, ..., 5-5.
Assuming you count 1-0 as different than 0-1.

I guess you would want the count for each possible scenario (probability of each).
Or maybe you count 0-1 with 5 shots per team different than 0-1 with 4 shots from the second team.
Your question is ambiguous, could you explain better?

[jay_shark]

For instance you may have something like wlwlwl_ _ _ _

Once the other team misses the 6th shot , it's over . There is no way for that team to come back . So the game can be settled in as few as 6 shots or at most 10 shots .

[Enrique]

[ QUOTE ]
For instance you may have something like wlwlwl_ _ _ _

Once the other team misses the 6th shot , it's over . There is no way for that team to come back . So the game can be settled in as few as 6 shots or at most 10 shots .

[/ QUOTE ]

Yes. I understand that.
But your question was ambiguous on defining what two "scoring scenarios are different"

From what you wrote there it seems that you name a counting scenario:
a_1b_1a_2b_2a_3b_3a_4b_4a_5b_5
where a_i are the shots of first team, b_i are the shots of the second team. (Of course you don't need the extra letters if the seventh shot is not necessary).

[jay_shark]

I didn't realize that it worked out to such a simple number .

The total number of scoring scenarios is just 10C6 =210 .

As a fun exercise , see if you can find a combinatorial argument to show that this is true .

[ispiked]

Can you show how you arrived at 10C6 as the answer, or is this the same question you're asking when you say "find a combinatorial argument..."?

[jay_shark]

Consider the numbers {1,2,3...10}.

Any subset of size 6 can uniquely represent a scoring scenario . If you select an odd number , then you've scored . If it is even number then your opponent missed . If you don't select an odd number , then you've missed . If you don't select an even number then your opponent scored .

For instance
{1,2,3,4,5,6} represents wlwlwl .
{1,2,3,4,5,7} represents wlwlwww