Combinatorics Selection

Question

There are N distinct boxes of balls in total. There are P boxes each containing A number of balls and remaining Q boxes contains B number of balls each.

Given a number X, what are the total the number of ways in which you can pick at least X balls from the boxes.

P+Q = N

Example: Number of P boxes=2 which contain 2 balls each, Number of Q boxes=1 which contain 2 balls. X=3(Given) where x=minimum number of balls to be picked So, P+Q=3 (total number of boxes) Combinations for the number of ways to pick atleast x i.e. 3 balls would be:

combinations of 3:(111),(210),(120),(021),(012),(201),(102)
combinations of 4:(220)(202)(022)(211)(121)(112)
combinations of 5:(212)(122)(221)
combinations of 6: (222)
total Combinations: 17

My Approach:
I have used "Stars and Bars Approach":

To calculate combinations of 6: x+y+z=6 which is converted into (2-x)+(2-y)+(2-z)=6 giving out x+y+z=0.
So, the combination of 6 becomes Binomial(2C2)=1
Similarly, Combinations of 5 becomes Binomial(3C2)=3
Combinations of 4= Binomial(4C2)=6
Combinations of 3= Binomial(5C2)=10

1+3+6+10=20
but the answer should be 1+3+6+7=17

Edge case has appeared on calculating the combinations of 3. How should I tackle this problem?

global total_combinations
total_combinations=0

from math import factorial

def combinations(a):
    global total_combinations
    bars=numberofAs+numberofBs-1
    stars=a
    total_combinations+=factorial(stars+bars)/(factorial(bars)*factorial(stars))


numberofAs,numberofBs,numberofballsinA,numberofballsinB=map(int,raw_input().split())

x=int(raw_input())

operational_array=[]

for i in range(numberofAs):
    operational_array.append(numberofballsinA)

for i in range(numberofBs):
    operational_array.append(numberofballsinB)

max_x=sum(operational_array) #calculate combinations from x to max_x
k=max_x

for i in range(max_x,x-1,-1):
    k=max_x-i
    combinations(k)

print total_combinations

PS:I found this question online but it seems interesting to me.