Best 4 out of 7
Here is a simple combinatorics problem to state. There was a somewhat similar, but much easier multiple choice question, on Brilliant.org so I must give credit to you guys for the inspiration. The problem I derived, on my own, is one that I decided is much more cool and difficult so I decided to give it a bash.
In a play-off series, the first team to win 4 games wins the series. Sometimes called a best 4 of 7.
What is the total number of different game sequences?
For example, in a 5 game series, WWLWW is one possible game sequence and in a 6 game series WLWLWW is one possible game sequence.
The answer is 35.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
1) Every game sequence in very case has to end in a W. For example:
4 game WWWW
5 Game WLWWW
6 game LWWLWW
7 game WLLWLWW
So we can drop the last W from each sequence. For example, in the 6 game sequences, drop the last W and we now have to place 2L’s in 5 slots.
_ L _ _ L Therefore 5C2 ways.
So altogether for the 4,5,6,7 cases we have 3C0 + 4C1 + 5C2 + 6C3 = 1 + 4 + 10 + 20 =35
2) Amazing! The 35 was immediately recognized by me as 7C3. It took me a few minutes to see how that could explain all 4 cases, but with the 7 and 3 being so obvious I just couldn’t let go.
You just have 7 slots to place 3L’s so 7C3. Now, scan each of the 35 arrangements from left to right until you get 4 W’s. Then you just throw away the rest and voila! you have all 35 in the 4 cases. BTW you can use a similar argument by placing the 4 W’s to get 7C4 = 35.
3) I thought about this a bit longer and it turns out that:
nC0 + (n+1)C1 + (n+2)C2 + (n+3)C3 = (n+4)C3
is an amazing identity in Pascal’s Triangle which I wasn’t aware of (I did prove it using factorials). This was icing on the cake. You can use it solve for “best of 3”, “best of 5”, ......