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Question 977381: Hi there tutors, I have no idea how to solve this, can u guys help me? Thanks
A hockey game between two teams is 'relatively close' if the number of goals scored by the two teams never differ by more than two. In how many ways can the first 12 goals of a game be scored if the game is 'relatively close'?
Answer by Fombitz(32388)   (Show Source):
You can put this solution on YOUR website! You're going to need a large sheet of paper and draw out all of the possible outcomes.
Let's call the two teams A and B.
Let's start by looking at the first goal, either A or B scores.
Then the second, again either A or B.
So the four possible outcomes are,
AA
AB
BA
BB
TO get the next set, just add an A and then a B to each previous outcome so for each previous outcomes, you'll get two new outcomes.
Now the third goal, no team can score three in a row or more so the possible outcomes are,
AAA <--- Not allowed
AAB
ABA
ABB
BAA
BAB
BBA
BBB <--- Not allowed
.
.
.
Fourth goal
AABA
AABB
ABAA
ABAB
ABBA
ABBB <--- Not allowed
BAAA <--- Not allowed
BAAB
BABA
BABB
BBAA
BBAB
.
.
Fifth goal
AABAA
AABAB
AABBA
AABBB <--- Not allowed
ABAAA <--- Not allowed
ABAAB
ABABA
ABABB
ABBAA
ABBAB
BAABA
BAABB
BABAA
BABAB
BABBA
BABBB <--- Not allowed
BBAAA <--- Not allowed
BBAAB
BBABA
BBABB
And so on until you get to 12 goals.
It's not difficult but it is tedious and you can easily make a mistake.
Just make sure you have twice the number of outcomes of the previous series before you start knocking out "Not allowed" goals to keep tabs on your work.
.
.
.
You can also generate all 2^12 (4096) possible outcomes and then go through and knockout the ones that have more than 2 in a row for either team. Six and 1/2 dozen the other.
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