Graphics and media/Collections of tips and tricks/Pascal's triangle

Pascal's Triangle is a triangula array of numbers which is well-known in mathematics. The number at the n-th row and r-th column in the array is the value of combination :

 {n \choose k} 

or

 \binom{n}{k} 

More detailed explanation, you may read wikipedia.

• Pascal's Triangle in Wikipedia

We draw Pascal's triangle using MetaFun and Lua. It can be done by the help of Hans.

First, we define a function to calculate the combination with Lua.

lua code for calculation

The functions MP.p_ncr(n,r) and MP.p_ncr_x() are provided for better and simpler code.

• \startluacode
    local function fact (n)
      if n <= 0 then
        return 1
      else
        return n * fact(n-1)
      end
    end
  
    local function ncr(n,r)
      return fact(n)/(fact(r)*fact(n-r))
    end
    userdata.P = {
         fact = fact,
         ncr  = ncr,
    }
    function MP.p_ncr(n, r)
      mp.print(ncr(n,r))
    end
    function MP.p_ncr_x()
      mp.print(ncr(mp.scan.pair()))
    end
  \stopluacode
  

• 

Drawing Pascal's triangle with lines

There are 3 different ways of calculating the value of combination tt.

• \startluacode
    local function fact (n)
      if n <= 0 then
        return 1
      else
        return n * fact(n-1)
      end
    end
  
    local function ncr(n,r)
      return fact(n)/(fact(r)*fact(n-r))
    end
    userdata.P = {
         fact = fact,
         ncr  = ncr,
    }
    function MP.p_ncr(n, r)
      mp.print(ncr(n,r))
    end
    function MP.p_ncr_x()
      mp.print(ncr(mp.scan.pair()))
    end
  \stopluacode
  
  \startMPcode
  numeric n,r,u,dx,dy,tt,final; u := 1.8cm;
  
  path p, q;
  pair A,now;
  
  A := dir(210)*u;
  dy := ypart A;
  dx := -2*(xpart A);
  
  for n=0 upto 4:
    for r=0 upto n: 
  %   tt := lua("mp.print(P.ncr(" & decimal n & "," &  decimal r & " ))");
  %   tt := lua.MP.p_ncr(n,r);
      tt := runscript("MP.p_ncr_x()") (n,r)  ;
      now := n*A + (r*dx,0);
      dotlabel(textext("\dm {\binom {" & decimal n & "}{" & decimal r & "} = " & decimal tt & "}"),now);
      draw (now+A) -- now -- (now+A+(dx,0));
    endfor;
  endfor;
  \stopMPcode
  

• 

Drawing Pascal's triangle of numbers

• \startluacode
    local function fact (n)
      if n <= 0 then
        return 1
      else
        return n * fact(n-1)
      end
    end
  
    local function ncr(n,r)
      return fact(n)/(fact(r)*fact(n-r))
    end
    userdata.P = {
         fact = fact,
         ncr  = ncr,
    }
    function MP.p_ncr(n, r)
      mp.print(ncr(n,r))
    end
    function MP.p_ncr_x()
      mp.print(ncr(mp.scan.pair()))
    end
  \stopluacode
  
  \startMPcode
  numeric n,r,u,dx,dy,tt,final; u := 1.2cm;
  
  path p, q;
  pair A,now;
  
  A := dir(210)*u;
  dy := ypart A;
  dx := -2*(xpart A);
  
  for n=0 upto 8:
    for r=0 upto n:
  %   tt := lua("mp.print(P.ncr(" & decimal n & "," &  decimal r & " ))");
  %   tt := lua.MP.p_ncr(n,r);
      tt := runscript("MP.p_ncr_x()") (n,r)  ;
      now := n*A + (r*dx,0);
      label(textext("$" & decimal tt & "$"),now);
     endfor;
  endfor;
  \stopMPcode
  

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