For binomial function nCr=k, given r and k find n

问题

I need a function that can solve the following: for a binomial function nCr=k, given r and k find n. in mathematics nCr=n!/r!(n-r)! I tried following but it doesn't solve it. for example 8C6=28, for my function the inputs are 6 and 28 and i want to find 8. This may not have exact integer number so I want to find an x>=n.

"""I am approaching it this way, i.e. find the solution of a polynomial function iteratively, hope there is a better way"""
def find_n(r,k):
    #solve_for_n_in(n*(n-1)...(n-r)=math.factorial(r)*k
    #in the above example solve_for_n(n*(n-1)(n-2)(n-3)(n-4)(n-5)=720*28)

    sum=math.factorial(r)*k
    n=r+1
    p=1

    while p<sum:
        p=1
        for i in range(0,r+2):
            p*=(n-i)
        n+=1
    return n-1

Thanks.

回答1:

I solved it the following way, i.e. find the solution of a polynomial function iteratively, hope there is a better way.

def find_n(r,k):
    #solve_for_n_in(n*(n-1)...(n-r)=math.factorial(r)*k
    #in the above example solve_for_n(n*(n-1)(n-2)(n-3)(n-4)(n-5)=720*28)

    target=math.factorial(r)*k
    n=r+1
    p=1

    while p<target:
        p=1
        for i in range(0,r+2):
            p*=(n-i)
        n+=1
    return n-1

回答2:

Here's a solution which uses fminsearch. You'll want to minimize the absolute difference between nchoosek(n, r) and k. However, you'll likely run into undefined values for nchoosek, so it's better to define it from scratch. Don't use factorial either though, as it's undefined for negative integers. Instead, use gamma (read about this on Wikipedia if you don't know).

r = 6;
k = 28;
toMinimize = @(n) abs(gamma(n+1) / (gamma(r+1) * gamma(n-r+1)) - k);

Be smart about the initial conditions:

for n = 1:10
    [res(n, 1), fval(n, 1)] = fminsearch(toMinimize, n);
end
[res fval]

Now you'll see you should only trust initial conditions n0 >= 5, for which the answer is n = 8.

ans =
             1.42626953125          27.9929874410369
             1.42626953125          27.9929874410369
           3.5737060546875          27.9929874410073
             3.57373046875          27.9929874410369
                         8                         0
          8.00000152587891       5.2032510172495e-05
          8.00000152587891      5.20325100552554e-05
                         8                         0
          7.99999694824218      0.000104064784270719
                         8                         0

来源：https://stackoverflow.com/questions/35753215/for-binomial-function-ncr-k-given-r-and-k-find-n