Question

import numpy as np

from scipy.special import comb

def cumulative_comb_with_repetition(n, k):

"""

Compute the number of possible non-negative, integer solutions to

x1 + x2 + ... + xk <= n.

  

We will use this function to compute the dimension

of order k polynomial feature vector

Args:

n: integer, the number of "balls" or "stars"

k: integer, the number of "urns" or "bars"

  

Returns: the total number of combinations, integer.

"""

# your code below

# your code above

raise NotImplementedError

import numpy as np from scipy.special import comb def cumulative_comb_with_repetition(n, k): Compute the number of possible non-negative, integer solutions to 1 + x2 + ... + xk <- n. 7 lution We will use this function to compute the dimension of order k polynomial feature vector Args : n: integer, the number of "balls" or "stars" k: integer, the number of "urns" or "bars" 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Returns: the total number of combinations, integer. # your code below your code above raise Not ImplementedError

The question doesn't ask for the number of one combination only but the total number of all combinations

Answer 1

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)C_k-1 when r.h.s=n

   The no of non negative solutions will be ^(n+k-2)C_k-1 when r.h.s=n-1

    The no of non negative solutions will be ^(n+k-3)C_k-1 when r.h.s=n-2

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

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

   The no of non negative solutions will be ^(k)C_k-1 when r.h.s=1

     The no of non negative solutions will be ^(k-1)C_k-1 when r.h.s=0

   So total no of non negative solutions will be

=^(k-1)C_k-1+ ^(k)C_k-1+ .................+^(n+k-3)C_k-1+^(n+k-2)C_k-1+^(n+k-1)C_k-1

=^(n+k)C_k (We know that ^(r)C_r+ ^(r+1)C_r+ .................+^(n-3)C_r+^(n-2)C_r+^(n-1)C_r+⁽ⁿ⁾C_r=⁽ⁿ⁺¹⁾C_(r+1).....combination formula)

Now we are going to code it out in python to find ^(n+k)C_k

python code snippet:>

In [1] : 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)) 56.0 In [2]: #getting the no of non negative solutions for xl+x2+x3+x4<=6 print (cumulative_comb_with_repetition (6,4)) 35.0

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.