9 dart outs

[ChrisWos]

There has been a thread today on the number of 9 dart finishes in a game of 501. With the rules being you must finish on a double there are 22 possible finishes but with each finish there are a number of paths (combinations).

out of curiosity I wanted to see if I could work out the formula or how to calculate the number of paths. But to no avail

any help appreciated

 

[stapel]

There has been a thread today on the number of 9 dart finishes in a game of 501.

A thread on this topic? Here?

Also, what is "a nine-dart finish"? What is "a game of 501"?

(We can try to assist with the mathematics, but you'll need to provide all of the other necessary information.)

With the rules being you must finish on a double there are 22 possible finishes but with each finish there are a number of paths (combinations).

out of curiosity I wanted to see if I could work out the formula or how to calculate the number of paths. But to no avail

What are your thoughts? What have you tried? Where are you stuck?

Also, what is the meaning of the codes (?) and numbers in the graphic?

Please be complete. Thank you!

Eliz.

 

[Otis]

It's been quite some time for me, but 501 is the name of the dart game where you start with 501 points and then subtract points as you throw darts, depending on where they land. The first player to subtract exactly 501 points in total wins. If a dart lands somewhere with no point value (like on the floor or in someone's neck), then you subtract zero for that throw. Different regions on the dart board have different point values (link below).

A 9-dart finish is to darts as 12 consecutive strikes are to bowling: a perfect score. ?

Darts - Wikipedia

en.wikipedia.org

Short video of first, televised 9-dart finish
[imath]\;[/imath]

 

[Otis]

must finish on a double there are 22 possible finishes but with each finish there are a number of paths (combinations)

Hi Chris. Looking at your chart, I think I understand it (except, possibly, for the specific point value of 'Bull').

It describes 9-dart finishes, so nine darts are thrown totaling 501 points.

Double-Out means the last dart lands in a region with doubled points.

The path counts suggest that you mean permutations instead of combinations. In combinations, order does not matter. When different orderings are counted separately, we call them permutations.

For example, the first row lists eight paths to reach 501 points when the last dart lands in the region for 12-doubled (D12). Seven darts must land in 20-tripled (T20) and one dart must land in 19-tripled (T19). What I just listed is a combination, so there is only one combination of 9 dart throws scoring 501 points when one of them scores 24 (D12).

Here are the eight permutations (i.e., different throwing orders possible), when the last throw scores 24:

T19 , T20 , T20 , T20 , T20 , T20 , T20 , T20 , D12
T20 , T19 , T20 , T20 , T20 , T20 , T20 , T20 , D12
T20 , T20 , T19 , T20 , T20 , T20 , T20 , T20 , D12
T20 , T20 , T20 , T19 , T20 , T20 , T20 , T20 , D12
T20 , T20 , T20 , T20 , T19 , T20 , T20 , T20 , D12
T20 , T20 , T20 , T20 , T20 , T19 , T20 , T20 , D12
T20 , T20 , T20 , T20 , T20 , T20 , T19 , T20 , D12
T20 , T20 , T20 , T20 , T20 , T20 , T20 , T19 , D12

Let me know, if I've misunderstood anything. Does your dart board have an inner bullseye (25 points) and outer bullseye (50 points)? If Bull=50 in the chart, then it doesn't seem to be doubled in the last throw.


[imath]\;[/imath]

 

[ChrisWos]

Hi Chris. Looking at your chart, I think I understand it (except, possibly, for the specific point value of 'Bull').

It describes 9-dart finishes, so nine darts are thrown totaling 501 points.

Double-Out means the last dart lands in a region with doubled points.

The path counts suggest that you mean permutations instead of combinations. In combinations, order does not matter. When different orderings are counted separately, we call them permutations.

For example, the first row lists eight paths to reach 501 points when the last dart lands in the region for 12-doubled (D12). Seven darts must land in 20-tripled (T20) and one dart must land in 19-tripled (T19). What I just listed is a combination, so there is only one combination of 9 dart throws scoring 501 points when one of them scores 24 (D12).

Here are the eight permutations (i.e., different throwing orders possible), when the last throw scores 24:

T19 , T20 , T20 , T20 , T20 , T20 , T20 , T20 , D12
T20 , T19 , T20 , T20 , T20 , T20 , T20 , T20 , D12
T20 , T20 , T19 , T20 , T20 , T20 , T20 , T20 , D12
T20 , T20 , T20 , T19 , T20 , T20 , T20 , T20 , D12
T20 , T20 , T20 , T20 , T19 , T20 , T20 , T20 , D12
T20 , T20 , T20 , T20 , T20 , T19 , T20 , T20 , D12
T20 , T20 , T20 , T20 , T20 , T20 , T19 , T20 , D12
T20 , T20 , T20 , T20 , T20 , T20 , T20 , T19 , D12

Let me know, if I've misunderstood anything. Does your dart board have an inner bullseye (25 points) and outer bullseye (50 points)? If Bull=50 in the chart, then it doesn't seem to be doubled in the last throw.


[imath]\;[/imath]

Yeah exactly that the bull has a value of 50 and when the questions was asked I thought the answer was simply 22. But what was being asked was how many permutations are there.
Where there are only 3 values ie T20 T19 and D12 then it’s quite simply 8 permutations but then anything outside that it makes a very long list.

I tried first N!/N-r but due to the fact that numbers can be repeated this doesn’t work

I then tried the formula that allows repetition but there’s a rule where a specific value has to be used a certain number of times ie on a D12 finish T20 must be used 7 times and D19 once this is where I’ve struggled to find the formula.

hope that makes sense

 

[Otis]

the bull has a value of 50

Very well; however, if a ninth dart hits the bullseye and 50 is not doubled to 100, then why is that game classified as a Double-Out finish?

I’ve struggled to find the formula

I think that we would need an algorithm, to generate the chart's permutation tallies.
[imath]\;[/imath]