Mathematical Combination

Question

Write a program that takes an input such as:

n,k

which then computes:

and then prints the result.

---

A numerical example:

Input:

5,2

Internal computation:

Printed Output:

10

---

I'd like to see an answer that beats my python solution of 65 characters, but all languages are obviously welcome.

Here's my solution:

n,k=input();f=lambda x:+(x<2)or x*f(x-1);print f(n)/(f(k)*f(n-k))

Edit:

I admit that this question is from the codegolf website mathematical combination puzzle. I know that my answer may look like not much progress can be made on it, but the leaders of this puzzle have solved it in nearly half as many characters.

The current lowest character counts by language are:

Perl: 35

Ruby: 36

Python: 39

PHP: 62

Answer 1

APL, 3 bytes

⎕!⎕

Or for those whose browser doesn't render the above, in an ASCII rendering:

{Quad}!{Quad}

Answer 2

R (11 Chars)

choose(n,r)

Answer 3

C 96

With I/O (which takes about 34 chars). Added a couple of newlines to make it readable.

main(a,b,c,d){scanf("%d,%d",&a,&b);
d=a-b;for(c=1;a>b;){c*=a--;}for(;d;)
{c/=d--;}printf("%d",c);}

Now if you'll excuse me, I have an ASCII n choose k rocket to catch.

    d;main(a
   ,         b
  ,           c
 )  int        a
 ;   {scanf    (
 (   "%d %d"   )
 ,   &a,  &b   )
 ;   d    =a   -
 b   +     b   -
 b   *     1   ;
 a             -
 a  ;for    (  c
 =  1;a>   b   ;
 )  {c=   c    *
 (  (a-- )     )
 ;  }for(      b
 =  b + 1      ;
 d  ;    )     {
 c  =     c    /
 (  d--    )   ;
  }           {
  }           {
   }         (
  printf("%d",c)
 )      ;       }
/*     *  *   * *\
 * * X   * 8 * * |
|     *      *    \
*/    //       *  */

Answer 4

GolfScript, 17 chars

This solution handles cases like k=0 or k=1 correctly.

~>.,,]{1\{)*}/}//

Factorial-like portion is based off a previous answer.

Answer 5

Ruby 1.9, 52 46 (42) characters

eval"A,B="+gets;i=0;p eval"(A-B+i+=1)/i*"*B+?1

If stderr is ignored:

eval"A,B=I="+gets;p eval"I/(A-I-=1)*"*B+?1

Ruby 1.8, 43 characters, no additional output to stderr:

eval"a,b=i="+gets;p eval"i/(a-i-=1)*"*b+"1"

Edits:

• (52 -> 48) Found a shorter way to parse the input
• (48 -> 46) Less looping, more eval.

Answer 6

GolfScript 21

~~)>.,,]{{)}%{*}*}%~/

Not particularly short, GolfScript lacks a real factorial function, however this has got to be the most wicked data manipulation that I have ever done, this calls for a stack trace:

"5,2" Data on the stack from input.
~ Eval command, note that , is an operator that turns a number into an array.
[0 1 2 3 4] 2
~ Binary not.
[0 1 2 3 4] -3
) Increment.
[0 1 2 3 4] -2
> Take end of array, -2 as parameter to get the last 2 elements.
[3 4]
. Duplicate element.
[3 4] [3 4]
, Array length.
[3 4] 2
, Turn number to array.
[3 4] [0 1]
] Create array.
[[3 4] [0 1]]
{{)}%{*}*} Block of code.
[[3 4] [0 1]] {{)}%{*}*}
% Execute block once for each element of the array. The following part only demonstrate the first loop.
[3 4]
{)}% Increment each array element.
[4 5]
{*} Block containing a multiply command.
[4 5] {*}
* "Fold" the array using the block command, that is in this case make the product of all elements.
20
After the big loop has finished it returns an array with the results.
[20 2]
~ Deconstruct the array.
20 2
/ Division.
10

Answer 7

Python (56)

f=lambda n,k:k<1and 1or f(n-1,k-1)*n/k;print f(*input())

Ungolfed code and some explanation of a shortcut for calculating the binomial coefficient. (Note: There is some insight that I just haven't figured out in order to get down to the 39 char version; I don't think this approach will get you there.)

# Since choose(n,k) =
#
#     n!/((n-k)!k!)
#
#          [n(n-1)...(n-k+1)][(n-k)...(1)]
#        = -------------------------------
#            [(n-k)...(1)][k(k-1)...(1)]
#
# We can cancel the terms:
#
#     [(n-k)...(1)]
#
# as they appear both on top and bottom, leaving:
#
#    n (n-1)     (n-k+1)
#    - ----- ... -------
#    k (k-1)       (1)
#
# which we might write as:
#
#      choose(n,k) = 1,                      if k = 0
#                  = (n/k)*choose(n-1, k-1), otherwise
#
def choose(n,k):
    if k < 1:
        return 1
    else:
        return choose(n-1, k-1) * n/k

# input() evaluates the string it reads from stdin, so "5,2" becomes
# (5,2) with no further action required on our part.
#
# In the golfed version, we make use of the `*` unpacking operator, 
# to unpack the tuple returned by input() directly into the arguments
# of f(), without the need for intermediate variables n, k at all.
#
n, k = input()

# This line is left as an exercise to the reader.
print choose(n, k)

Answer 8

RPL (4)

(using built-in function)

COMB

Answer 9

Windows PowerShell, 57

$a,$b=iex $input
$p=1
for($a-=$b;$a-ge1){$p*=1+$b/$a--}$p

Answer 10

J, 33 36

(":!~/".;._1}:toJ',',1!:1(3))1!:2(4)

35 characters are input, parsing and output. The other character, !, is n choose k.

I don't have Windows around for testing this at the moment, but I believe it should work there.

Answer 11

Q, 32 chars

{f:{1*/1.+(!)x};f[x]%f[y]*f x-y}

Answer 12

Perl 6 (55)

my ($a,$b)=lines;$_=1;for 1..$a-$b {$_+=$_*$b/$^a};.say

Answer 13

Perl 6, 25 16 bytes

-9 bytes thanks to nwellnhof

+*o&combinations

Try it online!

Anonymous function that takes two numbers and returns an int. This uses the built-in combinations and converts the returned list to an int.

Answer 14

RPL (22)

(not using built-in COMB function)

→ n k 'n!/(k!*(n-k)!)'

Answer 15

Q (50 45)

 f:{(prd 1.+til x)%((prd 1.+til y)*prd 1.+til x-y)}

You can shave a few characters off the above by removing redundant brackets and using 1*/ instead of prd.

f:{(1*/1.+til x)%(1*/1.+til y)*1*/1.+til x-y}

Answer 16

Mathematica 12

Straightforward, built-in function.

n~Binomial~k

Answer 17

PHP (71 79)

<?$a=fgetcsv(STDIN);$x=1;while($a[1]-$i)$x=$x*($a[0]-++$i+1)/$i;echo$x;

<?php $a=fgetcsv(STDIN);$x=1;while(++$i<=$a[1])$x=$x*($a[0]-$i+1)/$i;echo $x?>

Answer 18

Python (54)

f=lambda n,k:k<1or f(n-1,k-1)*n/k;print 1*f(*input())

Essentially the same as the Python one above, but I shave off four bytes by dropping the

and 1

from the function definition. However, this results in the function returning True instead of 1 if k=0, but this can be fixed by multiplying with 1 before printing, since 1*True=1, thus adding two bytes.

Answer 19

J, 11 characters

!~/".1!:1[1

Takes input from the keyboard.

    !~/".1!:1[1
5,2
10

Answer 20

Tcl, 80 bytes

proc C n\ k {proc f n {expr $n?($n)*\[f $n-1]:1}
expr [f $n]/([f $k]*[f $n-$k])}

Try it online!

Answer 21

Javascript, 27 bytes

First my own 35-byte solutions:

f=(n,k)=>n-k&&k?f(--n,k)+f(n,k-1):1

Or, alternatively,

f=(n,k,i=k)=>i?f(n-1,k-1,i-1)*n/k:1

The first working recursively, with the simple (n,k) = (n-1,k) + (n-1,k-1) rule. The second using that (n,k) = (n-1,k-1) * n/k.

EDIT

I just noticed the solution by Arnould in a duplicate of this:

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

Which is a whopping 8 bytes less (27 bytes)

Answer 22

TI-BASIC, 16 chars (8 bytes)

Ans(1) nCr Ans(2

Input is a list of length 2 in Ans.
Output is the result of the formula defined here.

If the above solution doesn't suffice, then the following 35 chars (24 bytes) solution also works:

Ans(2)!⁻¹(Ans(1)-Ans(2))!⁻¹Ans(1)!

---

Note: TI-BASIC is a tokenized language. Character count does not equal byte count.

Answer 23

Haskell (80)

f(_,0)=1
f(n,k)=n/k*f(n-1,k-1)
main=getLine>>=putStr.show.f.read.(++")").('(':)

But, if input in the format x y is allowed instead of in the format x,y, it's 74 chars:

f[_,0]=1
f[n,k]=n/k*f[n-1,k-1]
main=interact$show.f.map read.take 2.words

Answer 24

Scala 54

val n,k=readInt
((k+1)to n product)/(1 to(n-k)product)

Answer 25

Python (52)

 f=lambda n,k:k<1or f(n-1,k-1)*n/k;print+f(*input())

Improved from the other two by using print+ to convert the result of f from boolean to int in case k==0.

Still have no idea how to shrink it to 39, I wonder whether they are using lambda at all.

Answer 26

(The OP only loosely specified the input & output method/format, so the following seems acceptable.)

Sage Notebook (39 41 40)

In the current cell,

f=lambda n,k:k<1or f(n-1,k-1)*n/k;+f(*_)

where the input in the form n,k is entered & evaluated in the preceding cell. This simulates "command-line input" by assigning it to _ (similar to command-line arguments).

Sage Notebook (42 44 43)

Alternatively, using "in-source input" (with only the x= and newline characters adding to the score), e.g.,

x=5,2
f=lambda n,k:k<1or f(n-1,k-1)*n/k;+f(*x)

Both of these approaches are obviously spin-offs from earlier answers by others.