Using Solve function for polynomial

Hi there I want to solve an equation below but i faced a problem.

i tried to solve this with following code but it did not works:

i already have Fc and m , i tried to find nc Please Help

[John D'Errico]

What does this have to do with a polynomial?

Answer 1

[John D'Errico]

You cannot use solve to solve this because in general there is no exact solution. (And polynomials are still not involved.)

I suppose, if you want to use the extension of factorial into the real line using the special function gamma:

factorial(x) = gamma(x+1)

then you can write it as:

Fc - gamma(m+1)/(gamma(nc+1)*gamma(m-nc+1)) + nc == 0

but even here there will be no analytical solution.

syms nc
m = 5;
Fc = 8;

solve(Fc - gamma(m+1)/(gamma(nc+1)*gamma(m-nc+1))+nc,nc)
Warning: Cannot solve symbolically. Returning a numeric approximation instead. 
> In solve (line 303) 
ans =
2.7090797227280332212966072765956

I could have used fzero too there. Although there are many values of m for which fzero might fail due to numerical problems. Factorials get really large, really fast. But fzero will work here, using gamma:

fzero(@(nc) Fc - gamma(m+1)/(gamma(nc+1)*gamma(m-nc+1)) + nc,3)
ans =
          2.70907972272803

>> fzero(@(nc) Fc - gamma(m+1)/(gamma(nc+1)*gamma(m-nc+1)) + nc,1.5)
ans =
                         2

It turns out, there are multiple solutions to your problem.

[John D'Errico]

Sorry, I originally misread your question. I've now fixed my response.

But no. There are at least TWO solutions to the problem.

ezplot(@(nc) Fc - gamma(m+1)./(gamma(nc+1).*gamma(m-nc+1)) + nc)

grid on

2 is one solution. But 2.709... is the second solution, as you can see from the plot.

Answer 2

[Walter Roberson]

syms nc
solve(Fc == factorial(m) / (factorial(nc)*factorial(m-nc)) - nc)

[Alireza Lashgary]

thanks for your reply , but it did not work.

but it has a solution. Fc is 8 , m is 5 and nc must be 2 but i got that warning.