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if I have the equation $$f(x_1,x_2,x_3,x_4)=x_1x_3+x_2x_4+x_1x_2x_4+x_1x_2x_3x_4$$

  • How many equations are possible if two variables ?

when $x_1=x$ and the other four are y

when $x_2=x$ and the other four are y

when $x_3=x$ and the other four are y

when $x_4=x$ and the other four are y

I want to calculate the number of equations in each case

An example is if we have two variables. How many equation number we produce during compensation.

I suggest using combinations $C^2_4$ ... Is this true?

Can a program of work to calculate the number of possible cases if two variables

  • And if three variables ....3 variables: if one variable is x, another is y and the other two are z, the we have .... possible cases; if one variable is x, the other two are y, and the other z, then we have ....possible cases.

  • And the generalization of more than three.

For example: if I have 

f[a_, b_, c_, d_, e_] := a d + b e + b c d - a b c d - a b d e - b c d e + a b c d e

2 variables: if one variable is x, and the other four are y, we have five polynomials;

f[x, y, y, y, y]

f[y, x, y, y, y]

f[y, y, x, y, y]

f[y, y, y, x, y]

f[y, y, y, y, x]

Account Functions. How many function is there because order of variables is necessary

if two variables are x and the other three are y, we have $C^2 _5$ = 10 polynomials.

f[x, x, y, y, y]

f[x,y , x, y, y]

f[x, y, y, x, y]

f[x, y, y, y, x]

. . . . . . . . . . . . .

 f[y, x, x, y, y]

f[y,x , y, x, y]

f[y, x, y, y, x]

.....................

f[y,y , x, x, y]

f[y, y, x, y, x]

.......................

f[y, y, y, x, x]

Is this possible?

thanks for the help.

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  • $\begingroup$ "the other four" - but you only have four. $\endgroup$
    – J. M.'s persistent exhaustion ♦
    Apr 5, 2017 at 22:17
  • $\begingroup$ @J.M. Thanks for the note See the new update $\endgroup$
    – Emad kareem
    Apr 5, 2017 at 22:23

1 Answer 1

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I'm not quite sure what you need and I suspect your example is a reduced version of your actual problem, but a "brute force" approach is easily viable with only four parameters. 

f[a_, b_, c_, d_] := a c + b d + a b d + a b c d

f2[a : {_, _, _, _}] := f @@@ Permutations[a] // Union

f2[{x, y, y, y}] // Column

x y + y^2 + x y^2 + x y^3
x y + y^2 + y^3 + x y^3

f2[{x, y, z, z}] // Column

x z + y z + x y z + x y z^2
x y + x y z + z^2 + x y z^2
x z + y z + x z^2 + x y z^2
x y + z^2 + x z^2 + x y z^2
x z + y z + y z^2 + x y z^2
x y + z^2 + y z^2 + x y z^2

f2[{w, x, y, z}] // Length

12
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  • $\begingroup$ I want to calculate the number of equations in each case An example is if we have two variables. How many equation number we produce during compensation $\endgroup$
    – Emad kareem
    Apr 5, 2017 at 22:34
  • $\begingroup$ @Emadkareem Please edit your question to include explicit examples of what you mean. $\endgroup$
    – Mr.Wizard
    Apr 5, 2017 at 22:51
  • $\begingroup$ See the new update $\endgroup$
    – Emad kareem
    Apr 5, 2017 at 23:00
  • $\begingroup$ @Emad I don't understand. With your new example you seem to not want to eliminate equivalent polynomials (which is why I used Union) therefore I do not understand why the polynomial matters at all? Your values five and ten are directly computed with Multinomial[1, 4] and Multinomial[2, 3] -- is that all you want? $\endgroup$
    – Mr.Wizard
    Apr 5, 2017 at 23:04
  • $\begingroup$ See the new update ...In the first case of the example you will understand the idea $\endgroup$
    – Emad kareem
    Apr 5, 2017 at 23:17

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