>>13215695Assuming that order doesn't matter, your question can be described as: find the number of all the integer solutions to the following equation:
1 - a/2 - b/4 - c/8 - d/16 = 0
It's clear that a, b, c, and d can only take values in [0, 2], [0, 4], [0, 8] and [0, 16] respectively. I wrote a short python script that shows there are 35 such different solutions:
n = 0
for a in range(0, 3):
for b in range(0, 5):
for c in range(0, 9):
for d in range(0, 17):
res = 1 - a / 2 - b / 4 - c / 8 - d / 16
if res == 0:
n += 1
print("a = " + str(a) + ", b = " + str(b) + ", c = " + str(c) + ", d = " + str(d))
print(n)
This prints:
a = 0, b = 0, c = 0, d = 16
a = 0, b = 0, c = 1, d = 14
a = 0, b = 0, c = 2, d = 12
a = 0, b = 0, c = 3, d = 10
a = 0, b = 0, c = 4, d = 8
a = 0, b = 0, c = 5, d = 6
a = 0, b = 0, c = 6, d = 4
a = 0, b = 0, c = 7, d = 2
a = 0, b = 0, c = 8, d = 0
a = 0, b = 1, c = 0, d = 12
a = 0, b = 1, c = 1, d = 10
a = 0, b = 1, c = 2, d = 8
a = 0, b = 1, c = 3, d = 6
a = 0, b = 1, c = 4, d = 4
a = 0, b = 1, c = 5, d = 2
a = 0, b = 1, c = 6, d = 0
a = 0, b = 2, c = 0, d = 8
a = 0, b = 2, c = 1, d = 6
a = 0, b = 2, c = 2, d = 4
a = 0, b = 2, c = 3, d = 2
a = 0, b = 2, c = 4, d = 0
a = 0, b = 3, c = 0, d = 4
a = 0, b = 3, c = 1, d = 2
a = 0, b = 3, c = 2, d = 0
a = 0, b = 4, c = 0, d = 0
a = 1, b = 0, c = 0, d = 8
a = 1, b = 0, c = 1, d = 6
a = 1, b = 0, c = 2, d = 4
a = 1, b = 0, c = 3, d = 2
a = 1, b = 0, c = 4, d = 0
a = 1, b = 1, c = 0, d = 4
a = 1, b = 1, c = 1, d = 2
a = 1, b = 1, c = 2, d = 0
a = 1, b = 2, c = 0, d = 0
a = 2, b = 0, c = 0, d = 0
35
How many permutations of 35 take 4 we have? Well, we have to check each case, ignore the parameters that have zero, and permute the rest, i.e.:
perms = 0
for t in solutions:
nonzeroCount = 0
for i in t:
if i != 0: nonzeroCount += 1
perms += factorial(nonzeroCount)
print(perms)
which prints:
144
