Different combinations possible

Clash Royale CLAN TAG#URR8PPP

up vote
4
down vote

favorite

Problem

Given a value n, imagine a mountain landscape inscribed in a reference (0, 0) to (2n, 0).
There musn't be white spaces between slopes and also the mountain musn't descend below the x axis.
The problem to be solved is: given n (which defines the size of the landscape) and the number k of peaks
(k always less than or equal to n), how many combinations of mountains are possible with k peaks?

Input

n who represents the width of the landscape and k which is the number of peaks.

Output

Just the number of combinations possible.

Example

Given n=3 and k=2 the answer is 3 combinations.

Just to give a visual example, they are the following:

 / / //
// / / / 

are the 3 combinations possible using 6 (3*2) positions and 2 peaks.

Edit:
- more examples -

n k result
2 1 1
4 1 1
4 3 6
5 2 10

code-golf combinatorics recursion

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edited 1 hour ago

dylnan

3,8482528

asked 2 hours ago

combinationsD

212

New contributor

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Is this the same as, "find the number of expressions of n matched parentheses pairs that contain exactly k instances of ()"?
  – xnor
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  https://oeis.org/A001263?
  – xnor
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  This also needs some more test cases for us to verify our solutions against.
  – Shaggy
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @xnor yes it is.
  – Jonathan Allan
  41 mins ago
  

  
  
  
  

add a comment | 

up vote
4
down vote

favorite

Problem

Given a value n, imagine a mountain landscape inscribed in a reference (0, 0) to (2n, 0).
There musn't be white spaces between slopes and also the mountain musn't descend below the x axis.
The problem to be solved is: given n (which defines the size of the landscape) and the number k of peaks
(k always less than or equal to n), how many combinations of mountains are possible with k peaks?

Input

n who represents the width of the landscape and k which is the number of peaks.

Output

Just the number of combinations possible.

Example

Given n=3 and k=2 the answer is 3 combinations.

Just to give a visual example, they are the following:

 / / //
// / / / 

are the 3 combinations possible using 6 (3*2) positions and 2 peaks.

Edit:
- more examples -

n k result
2 1 1
4 1 1
4 3 6
5 2 10

code-golf combinatorics recursion

share|improve this question

edited 1 hour ago

dylnan

3,8482528

asked 2 hours ago

combinationsD

212

New contributor

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Is this the same as, "find the number of expressions of n matched parentheses pairs that contain exactly k instances of ()"?
  – xnor
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  https://oeis.org/A001263?
  – xnor
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  This also needs some more test cases for us to verify our solutions against.
  – Shaggy
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @xnor yes it is.
  – Jonathan Allan
  41 mins ago
  

  
  
  
  

add a comment | 

up vote
4
down vote

favorite

up vote
4
down vote

favorite

Problem

Given a value n, imagine a mountain landscape inscribed in a reference (0, 0) to (2n, 0).
There musn't be white spaces between slopes and also the mountain musn't descend below the x axis.
The problem to be solved is: given n (which defines the size of the landscape) and the number k of peaks
(k always less than or equal to n), how many combinations of mountains are possible with k peaks?

Input

n who represents the width of the landscape and k which is the number of peaks.

Output

Just the number of combinations possible.

Example

Given n=3 and k=2 the answer is 3 combinations.

Just to give a visual example, they are the following:

 / / //
// / / / 

are the 3 combinations possible using 6 (3*2) positions and 2 peaks.

Edit:
- more examples -

n k result
2 1 1
4 1 1
4 3 6
5 2 10

code-golf combinatorics recursion

share|improve this question

edited 1 hour ago

dylnan

3,8482528

asked 2 hours ago

combinationsD

212

New contributor

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

Problem

Given a value n, imagine a mountain landscape inscribed in a reference (0, 0) to (2n, 0).
There musn't be white spaces between slopes and also the mountain musn't descend below the x axis.
The problem to be solved is: given n (which defines the size of the landscape) and the number k of peaks
(k always less than or equal to n), how many combinations of mountains are possible with k peaks?

Input

n who represents the width of the landscape and k which is the number of peaks.

Output

Just the number of combinations possible.

Example

Given n=3 and k=2 the answer is 3 combinations.

Just to give a visual example, they are the following:

 / / //
// / / / 

are the 3 combinations possible using 6 (3*2) positions and 2 peaks.

Edit:
- more examples -

n k result
2 1 1
4 1 1
4 3 6
5 2 10

code-golf combinatorics recursion

code-golf combinatorics recursion

share|improve this question

edited 1 hour ago

dylnan

3,8482528

asked 2 hours ago

combinationsD

212

New contributor

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|improve this question

edited 1 hour ago

dylnan

3,8482528

asked 2 hours ago

combinationsD

212

New contributor

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|improve this question

share|improve this question

edited 1 hour ago

dylnan

3,8482528

edited 1 hour ago

dylnan

3,8482528

edited 1 hour ago

dylnan

3,8482528

3,8482528

asked 2 hours ago

combinationsD

212

New contributor

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 2 hours ago

combinationsD

212

asked 2 hours ago

combinationsD

212

212

New contributor

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

New contributor

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

combinationsD is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Is this the same as, "find the number of expressions of n matched parentheses pairs that contain exactly k instances of ()"?
  – xnor
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  https://oeis.org/A001263?
  – xnor
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  This also needs some more test cases for us to verify our solutions against.
  – Shaggy
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @xnor yes it is.
  – Jonathan Allan
  41 mins ago
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Is this the same as, "find the number of expressions of n matched parentheses pairs that contain exactly k instances of ()"?
  – xnor
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  https://oeis.org/A001263?
  – xnor
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  This also needs some more test cases for us to verify our solutions against.
  – Shaggy
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @xnor yes it is.
  – Jonathan Allan
  41 mins ago
  

  
  
  
  

2

2

Is this the same as, "find the number of expressions of n matched parentheses pairs that contain exactly k instances of ()"?
– xnor
1 hour ago

Is this the same as, "find the number of expressions of n matched parentheses pairs that contain exactly k instances of ()"?
– xnor
1 hour ago

1

1

https://oeis.org/A001263?
– xnor
1 hour ago

https://oeis.org/A001263?
– xnor
1 hour ago

1

1

This also needs some more test cases for us to verify our solutions against.
– Shaggy
1 hour ago

This also needs some more test cases for us to verify our solutions against.
– Shaggy
1 hour ago

@xnor yes it is.
– Jonathan Allan
41 mins ago

@xnor yes it is.
– Jonathan Allan
41 mins ago

add a comment | 

4 Answers
4

active

oldest

votes

up vote
0
down vote

Jelly, 8 bytes

,’$c’}P:

A dyadic Link accepting n on the left and k on the right which yields the count.

Try it online!

share|improve this answer

answered 42 mins ago

Jonathan Allan

48.6k534160

add a comment | 

up vote
0
down vote

APL(Dyalog), 19 18 16 bytes

⍺÷⍨×/(⍵,⍵-1)!⍺

Uses the identity in the OEIS sequence. Takes n on the left and k on the right.

TIO

share|improve this answer

answered 30 mins ago

Quintec

663315

add a comment | 

up vote
0
down vote

Jelly, 7 bytes

cⱮṫ-P÷⁸

Try it online!

Takes input as n then k. Uses the formula

$N(n,k)=frac1nbinomnkbinomnk-1$

which I found on Wikipedia.

cⱮṫ-P÷⁸
c Binomial coefficient of n and...
 â±® each of 1..k
 ṫ- Keep the last two. ṫ is tail, - is -1.
 P Product of the two numbers.
 ÷ Divide by
 ⁸ n.

7 bytes

Each line works on it's own.

,’$c@P÷
c@€ṫ-P÷

Takes input as k then n.

share|improve this answer

edited 21 mins ago

answered 40 mins ago

dylnan

3,8482528

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Wait... tail is automatically defined as 2 numbers? (Don't know Jelly at all, just a silly question)
  – Quintec
  20 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Quintec There are two tail functions. One (Ṫ) that just takes the last element of a single argument and the one I used (ṫ) which takes two arguments. The fist argument is a list and the second one is a number (In my case -1 represented by a - in the code) which tells you how many elements to save. Having -1 give two elements was the golfiest way to define ṫ
  – dylnan
  15 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Gotcha, thanks! I see how jelly was built for golf... hehe
  – Quintec
  12 mins ago
  

  
  
  
  

add a comment | 

up vote
0
down vote

JavaScript (ES7), 59 58 49 bytes

Takes input as (n)(k).

n=>k=>(g=k=>k?m--*g(k-1)/k:1)(k,m=n)*g(k-1,m=n)/n

Try it online!

Computes:

$$a_n,k=frac1kbinomn-1k-1binomnk-1=frac1nbinomnkbinomnk-1$$

This formula comes from A001263, the OEIS sequence that xnor referred to in the challenge comments.

share|improve this answer

edited 21 mins ago

answered 41 mins ago

Arnauld

65.5k582277

add a comment | 

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4 Answers
4

active

oldest

votes

4 Answers
4

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

Jelly, 8 bytes

,’$c’}P:

A dyadic Link accepting n on the left and k on the right which yields the count.

Try it online!

share|improve this answer

answered 42 mins ago

Jonathan Allan

48.6k534160

add a comment | 

up vote
0
down vote

Jelly, 8 bytes

,’$c’}P:

A dyadic Link accepting n on the left and k on the right which yields the count.

Try it online!

share|improve this answer

answered 42 mins ago

Jonathan Allan

48.6k534160

add a comment | 

up vote
0
down vote

up vote
0
down vote

Jelly, 8 bytes

,’$c’}P:

A dyadic Link accepting n on the left and k on the right which yields the count.

Try it online!

share|improve this answer

answered 42 mins ago

Jonathan Allan

48.6k534160

Jelly, 8 bytes

,’$c’}P:

A dyadic Link accepting n on the left and k on the right which yields the count.

Try it online!

share|improve this answer

answered 42 mins ago

Jonathan Allan

48.6k534160

share|improve this answer

share|improve this answer

answered 42 mins ago

Jonathan Allan

48.6k534160

answered 42 mins ago

Jonathan Allan

48.6k534160

answered 42 mins ago

Jonathan Allan

48.6k534160

48.6k534160

add a comment | 

add a comment | 

up vote
0
down vote

APL(Dyalog), 19 18 16 bytes

⍺÷⍨×/(⍵,⍵-1)!⍺

Uses the identity in the OEIS sequence. Takes n on the left and k on the right.

TIO

share|improve this answer

answered 30 mins ago

Quintec

663315

add a comment | 

up vote
0
down vote

APL(Dyalog), 19 18 16 bytes

⍺÷⍨×/(⍵,⍵-1)!⍺

Uses the identity in the OEIS sequence. Takes n on the left and k on the right.

TIO

share|improve this answer

answered 30 mins ago

Quintec

663315

add a comment | 

up vote
0
down vote

up vote
0
down vote

APL(Dyalog), 19 18 16 bytes

⍺÷⍨×/(⍵,⍵-1)!⍺

Uses the identity in the OEIS sequence. Takes n on the left and k on the right.

TIO

share|improve this answer

answered 30 mins ago

Quintec

663315

APL(Dyalog), 19 18 16 bytes

⍺÷⍨×/(⍵,⍵-1)!⍺

Uses the identity in the OEIS sequence. Takes n on the left and k on the right.

TIO

share|improve this answer

answered 30 mins ago

Quintec

663315

share|improve this answer

share|improve this answer

answered 30 mins ago

Quintec

663315

answered 30 mins ago

Quintec

663315

answered 30 mins ago

Quintec

663315

663315

add a comment | 

add a comment | 

up vote
0
down vote

Jelly, 7 bytes

cⱮṫ-P÷⁸

Try it online!

Takes input as n then k. Uses the formula

$N(n,k)=frac1nbinomnkbinomnk-1$

which I found on Wikipedia.

cⱮṫ-P÷⁸
c Binomial coefficient of n and...
 â±® each of 1..k
 ṫ- Keep the last two. ṫ is tail, - is -1.
 P Product of the two numbers.
 ÷ Divide by
 ⁸ n.

7 bytes

Each line works on it's own.

,’$c@P÷
c@€ṫ-P÷

Takes input as k then n.

share|improve this answer

edited 21 mins ago

answered 40 mins ago

dylnan

3,8482528

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Wait... tail is automatically defined as 2 numbers? (Don't know Jelly at all, just a silly question)
  – Quintec
  20 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Quintec There are two tail functions. One (Ṫ) that just takes the last element of a single argument and the one I used (ṫ) which takes two arguments. The fist argument is a list and the second one is a number (In my case -1 represented by a - in the code) which tells you how many elements to save. Having -1 give two elements was the golfiest way to define ṫ
  – dylnan
  15 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Gotcha, thanks! I see how jelly was built for golf... hehe
  – Quintec
  12 mins ago
  

  
  
  
  

add a comment | 

up vote
0
down vote

Jelly, 7 bytes

cⱮṫ-P÷⁸

Try it online!

Takes input as n then k. Uses the formula

$N(n,k)=frac1nbinomnkbinomnk-1$

which I found on Wikipedia.

cⱮṫ-P÷⁸
c Binomial coefficient of n and...
 â±® each of 1..k
 ṫ- Keep the last two. ṫ is tail, - is -1.
 P Product of the two numbers.
 ÷ Divide by
 ⁸ n.

7 bytes

Each line works on it's own.

,’$c@P÷
c@€ṫ-P÷

Takes input as k then n.

share|improve this answer

edited 21 mins ago

answered 40 mins ago

dylnan

3,8482528

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Wait... tail is automatically defined as 2 numbers? (Don't know Jelly at all, just a silly question)
  – Quintec
  20 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Quintec There are two tail functions. One (Ṫ) that just takes the last element of a single argument and the one I used (ṫ) which takes two arguments. The fist argument is a list and the second one is a number (In my case -1 represented by a - in the code) which tells you how many elements to save. Having -1 give two elements was the golfiest way to define ṫ
  – dylnan
  15 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Gotcha, thanks! I see how jelly was built for golf... hehe
  – Quintec
  12 mins ago
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

Jelly, 7 bytes

cⱮṫ-P÷⁸

Try it online!

Takes input as n then k. Uses the formula

$N(n,k)=frac1nbinomnkbinomnk-1$

which I found on Wikipedia.

cⱮṫ-P÷⁸
c Binomial coefficient of n and...
 â±® each of 1..k
 ṫ- Keep the last two. ṫ is tail, - is -1.
 P Product of the two numbers.
 ÷ Divide by
 ⁸ n.

7 bytes

Each line works on it's own.

,’$c@P÷
c@€ṫ-P÷

Takes input as k then n.

share|improve this answer

edited 21 mins ago

answered 40 mins ago

dylnan

3,8482528

Jelly, 7 bytes

cⱮṫ-P÷⁸

Try it online!

Takes input as n then k. Uses the formula

$N(n,k)=frac1nbinomnkbinomnk-1$

which I found on Wikipedia.

cⱮṫ-P÷⁸
c Binomial coefficient of n and...
 â±® each of 1..k
 ṫ- Keep the last two. ṫ is tail, - is -1.
 P Product of the two numbers.
 ÷ Divide by
 ⁸ n.

7 bytes

Each line works on it's own.

,’$c@P÷
c@€ṫ-P÷

Takes input as k then n.

share|improve this answer

edited 21 mins ago

answered 40 mins ago

dylnan

3,8482528

share|improve this answer

share|improve this answer

edited 21 mins ago

edited 21 mins ago

edited 21 mins ago

answered 40 mins ago

dylnan

3,8482528

answered 40 mins ago

dylnan

3,8482528

answered 40 mins ago

dylnan

3,8482528

3,8482528

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Wait... tail is automatically defined as 2 numbers? (Don't know Jelly at all, just a silly question)
  – Quintec
  20 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Quintec There are two tail functions. One (Ṫ) that just takes the last element of a single argument and the one I used (ṫ) which takes two arguments. The fist argument is a list and the second one is a number (In my case -1 represented by a - in the code) which tells you how many elements to save. Having -1 give two elements was the golfiest way to define ṫ
  – dylnan
  15 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Gotcha, thanks! I see how jelly was built for golf... hehe
  – Quintec
  12 mins ago
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Wait... tail is automatically defined as 2 numbers? (Don't know Jelly at all, just a silly question)
  – Quintec
  20 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Quintec There are two tail functions. One (Ṫ) that just takes the last element of a single argument and the one I used (ṫ) which takes two arguments. The fist argument is a list and the second one is a number (In my case -1 represented by a - in the code) which tells you how many elements to save. Having -1 give two elements was the golfiest way to define ṫ
  – dylnan
  15 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Gotcha, thanks! I see how jelly was built for golf... hehe
  – Quintec
  12 mins ago
  

  
  
  
  

Wait... tail is automatically defined as 2 numbers? (Don't know Jelly at all, just a silly question)
– Quintec
20 mins ago

Wait... tail is automatically defined as 2 numbers? (Don't know Jelly at all, just a silly question)
– Quintec
20 mins ago

@Quintec There are two tail functions. One (Ṫ) that just takes the last element of a single argument and the one I used (ṫ) which takes two arguments. The fist argument is a list and the second one is a number (In my case -1 represented by a - in the code) which tells you how many elements to save. Having -1 give two elements was the golfiest way to define ṫ
– dylnan
15 mins ago

@Quintec There are two tail functions. One (Ṫ) that just takes the last element of a single argument and the one I used (ṫ) which takes two arguments. The fist argument is a list and the second one is a number (In my case -1 represented by a - in the code) which tells you how many elements to save. Having -1 give two elements was the golfiest way to define ṫ
– dylnan
15 mins ago

Gotcha, thanks! I see how jelly was built for golf... hehe
– Quintec
12 mins ago

Gotcha, thanks! I see how jelly was built for golf... hehe
– Quintec
12 mins ago

add a comment | 

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0
down vote

JavaScript (ES7), 59 58 49 bytes

Takes input as (n)(k).

n=>k=>(g=k=>k?m--*g(k-1)/k:1)(k,m=n)*g(k-1,m=n)/n

Try it online!

Computes:

$$a_n,k=frac1kbinomn-1k-1binomnk-1=frac1nbinomnkbinomnk-1$$

This formula comes from A001263, the OEIS sequence that xnor referred to in the challenge comments.

share|improve this answer

edited 21 mins ago

answered 41 mins ago

Arnauld

65.5k582277

add a comment | 

up vote
0
down vote

JavaScript (ES7), 59 58 49 bytes

Takes input as (n)(k).

n=>k=>(g=k=>k?m--*g(k-1)/k:1)(k,m=n)*g(k-1,m=n)/n

Try it online!

Computes:

$$a_n,k=frac1kbinomn-1k-1binomnk-1=frac1nbinomnkbinomnk-1$$

This formula comes from A001263, the OEIS sequence that xnor referred to in the challenge comments.

share|improve this answer

edited 21 mins ago

answered 41 mins ago

Arnauld

65.5k582277

add a comment | 

up vote
0
down vote

up vote
0
down vote

JavaScript (ES7), 59 58 49 bytes

Takes input as (n)(k).

n=>k=>(g=k=>k?m--*g(k-1)/k:1)(k,m=n)*g(k-1,m=n)/n

Try it online!

Computes:

$$a_n,k=frac1kbinomn-1k-1binomnk-1=frac1nbinomnkbinomnk-1$$

This formula comes from A001263, the OEIS sequence that xnor referred to in the challenge comments.

share|improve this answer

edited 21 mins ago

answered 41 mins ago

Arnauld

65.5k582277

JavaScript (ES7), 59 58 49 bytes

Takes input as (n)(k).

n=>k=>(g=k=>k?m--*g(k-1)/k:1)(k,m=n)*g(k-1,m=n)/n

Try it online!

Computes:

$$a_n,k=frac1kbinomn-1k-1binomnk-1=frac1nbinomnkbinomnk-1$$

This formula comes from A001263, the OEIS sequence that xnor referred to in the challenge comments.

share|improve this answer

edited 21 mins ago

answered 41 mins ago

Arnauld

65.5k582277

share|improve this answer

share|improve this answer

edited 21 mins ago

edited 21 mins ago

edited 21 mins ago

answered 41 mins ago

Arnauld

65.5k582277

answered 41 mins ago

Arnauld

65.5k582277

answered 41 mins ago

Arnauld

65.5k582277

65.5k582277

add a comment | 

add a comment | 

combinationsD is a new contributor. Be nice, and check out our Code of Conduct.

 

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combinationsD is a new contributor. Be nice, and check out our Code of Conduct.

combinationsD is a new contributor. Be nice, and check out our Code of Conduct.

combinationsD is a new contributor. Be nice, and check out our Code of Conduct.

 

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