Finding number of bounded premuations

bound_premuations.py

import math

def nCr(n,r):
    f = math.factorial
    return f(n) / f(r) / f(n-r)

def itemCount_cal(target, items, minValue):
    return target- items*minValue

def distributions(itemCount, bucketCount):
    #get all possible solutions
    f = math.factorial
    return f(itemCount)/(f(itemCount-bucketCount) *  f(bucketCount))
    

if __name__ == '__main__':
    target_range = 100
    bucketCount = 4
    lowerLimit = 20
    upperLimit = 60

   
    itemCount = itemCount_cal(target_range,bucketCount,lowerLimit)
    a = bucketCount * distributions((upperLimit+1) -itemCount, bucketCount)
    b = nCr(bucketCount,2) * distributions(2*(upperLimit+1)-itemCount, bucketCount)
    c = nCr(bucketCount,3) * distributions(3*(upperLimit+1)-itemCount, bucketCount)
    print c - a - b