venkman@bebop:~$ bc     <----- Start up the bc calculator program
                                    on cowboy.cns.uni.edu computer.
define f (n) {                         ------------------
   if (n <= 1) return (1);
   return (n * f(n - 1));
}

f(3)                 <----- Use the function f() with the value 3, 
6                              to calculate 3 factorial.

f(5)                 <-- f(5) outputs 120
120                
                         f is the program, 5 is the input, 120 is result
define p(n, r) {
   a = 1;
   for (i = n; i >= (n - r + 1); i--)
      a = a * i;
   return (a);
}

p(10,3)
720

p(6,3)
120 
         
define c(n, r) {
   return ( p(n, r) / f(r) );
}
     
c(10, 3)
120

c(6, 3)
20

c(10, 2)
45

^d   Ctrl+d  (Hold down the Control key, then type letter d)
          (Ctrl+d is signals the bc calculator you are done)

----------------------------------------------------------------
In the bc calculator: operators are:
        + for addition, - for subtract, 
        * for multiply, / for division and ^ for exponentiation
Examples:
2*2*2*2*2*2*2*2
256

2^8
256

scale=2
(26*25*24*23*22*21) / (26^6)
.53

There are three special variables, scale, ibase, and obase:

         scale defines how some operations use digits after
         the decimal point.  The default value of scale is 0. 
scale=0
(10*9*8*7)/(10^4)
0

scale=3
(10*9*8*7)/(10^4)
.504          

  ibase and obase define the conversion base for input and  output
  numbers.  The  default  for both input and output is base 10.  

obase=2       <---- Change the output base to binary, base 2
15
1111
255
11111111      <---- 255 in base 10 is 11111111 in binary base 2
512
1000000000    <---- 512 is 2 ^ 9, 2 to the 9th power

obase=16
15
F             <---- 16 digits in base 16 hexadecimal 
255                 Use A for 10, B for 11, C for 12, ..., F for 15
FF                  The letters A through F are used for the highest
512                 six digits of the sixteen digits.
200

obase=10      <---- Output base (obase) is set to decimal, base 10
ibase=2       <---- Input base  (ibase) is set to base 2
11111110
254
10000
16
100001
33    


----------------------------------------------------------------
for loops in the bc calculator
----------------------------------------------------------------
for  (  [expression1]  ;  [expression2]  ; [expression3] )
   statement

   The  for  statement  controls repeated execution of
   the statement.  Expression1 is evaluated before the
   loop.   Expression2 is evaluated before each execu­
   tion of the statement.   If  it  is  non-zero,  the
   statement is evaluated.  If it is zero, the loop is
   terminated.  After each execution of the statement,
   expression3 is evaluated before the reevaluation of
   expression2.  

define p(n, r) {
   a = 1;
   for (i = n; i >= (n - r + 1); i--)
      a = a * i;
   return (a);
}
                  expression1 is -->  i = n;
                  expression2 is -->  i >= (n - r + 1);
                  expression3 is -->  i--;

Here is an example of using variables in the bc calculator:

rad = 3
pi = 3.14159
area = pi * rad ^ 2
area
28.27431   

define a (r) {                   <--- Define a function named a()
   return ( 3.14159 * r ^ 2 );        to calculate area of circle.
}

a(3)
28.27431

a(10)
314.15900 
--------------------------------------------------------------


Exercise:  Using only the *, / and possibily the ^ operators,
           write down the expression that could be typed into 
           the bc calculator to calculate how many years it
           would take to solve the following problem by brute
           force on a computer:

           There are 20 different pilots available to be 
           assigned to 20 different flights.  The flights
           differ as to type of aircraft and what time of
           day and day they occur on and what destination
           city or base and originating city or base they
           have.

           The pilots each have expressed their preferences
           from highest to lowest.  Each airline or some 
           management team for the airline or airlines has
           listed which pilot is most preferred, 2nd most 
           preferred, down to least preferred.

           How many ways can the flights be arranged and pilots
           assigned or matched to them?

           Suppose a computer is capable of examining 1,000,000
           of the possible flight assignments each second.
           It can check 1 million arrangements per second and
           keep track of the best scoring arrangement.

           How many years would it take for this computer to
           examine all of the flight assignments and when done
           display the best possible matching of pilots to flights?
---------------------------------------------------------------------


Alternative bc definitions of factorial function:

     define f(n) {                           
         a = 1;                                  

         for (i = 1; i <= n; i++)                
            a = a * i;                              

         return (a);                             
     }                                       	
	
define f(n) {
    a = 1;

    for (i = n; i > 1; i--)
       a = a * i;

    return (a);
}