part1:
the given inequality is x1+x2+x3+........+xk<=n
we can say that x1+x2+x3+....+xk can be n,n-1,n-2,........,3,2,1,0 according to the given inequality.
The no of non negative solutions will be (n+k-1)Ck-1 when r.h.s=n
The no of non negative solutions will be (n+k-2)Ck-1 when r.h.s=n-1
The no of non negative solutions will be (n+k-3)Ck-1 when r.h.s=n-2
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The no of non negative solutions will be (k)Ck-1 when r.h.s=1
The no of non negative solutions will be (k-1)Ck-1 when r.h.s=0
So total no of non negative solutions will be
=(k-1)Ck-1+ (k)Ck-1+ .................+(n+k-3)Ck-1+(n+k-2)Ck-1+(n+k-1)Ck-1
=(n+k)Ck (We know that (r)Cr+ (r+1)Cr+ .................+(n-3)Cr+(n-2)Cr+(n-1)Cr+(n)Cr=(n+1)C(r+1).....combination formula)
Now we are going to code it out in python to find (n+k)Ck
python code snippet:>
text code:>
import numpy as np
from scipy.special import comb
def cumulative_comb_with_repetition(n,k):
#using comb() function to calculate
(n+k)C(k)
return comb(n+k,k)
#getting the no of non negative solutions for x1+x2+x3<=5
print(cumulative_comb_with_repetition(5,3))
#getting the no of non negative solutions for
x1+x2+x3+x4<=6
print(cumulative_comb_with_repetition(6,4))
output:>
56.0
35.0
Hope the solution is useful.If you have any problem regarding the answer,then comment down below.
import numpy as np from scipy.special import comb def cumulative_comb_with_repetition(n, k): """ Compute the number of...
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