combinations with repetitions

'''
Assume that we have a set A with n elements. Any selection of r objects from A, where each object can be selected more than 
once, is called a combination of n objects taken r at a time with repetition.
'''

#A bakery boasts eight varieties of doughnuts. If a box of doughnuts contains one dozen, 
#how many different options are there for a box of doughnuts?
# ans : (r+k -1)C(r) 



q = []
a = np.arange(1,9) #types of doughnuts = k = 8
l = a.tolist()
r = 12  # how many doughnuts we need in one box, r>k

random.shuffle(a)
b = a[:(8)]
random.shuffle(a)
d = a[:3]
print(d)
d = d.tolist()
b = b.tolist()
c = l + b  #join both the lists 


c+d #--> this is a multiset, where since 12>8, here len(c+d) = r=k-1 = 19, since in the end we need to calculate 19C12 

#duplicates of any of the 8 available
p = list(combinations(c+d,r)) #diff 1 dozen combos of 8 doughnut varieties
len(p)