Solution:

The given system of equations can be written as:

a + b + c + d + e = 20 ...(i)
a + b + c = 5 ...(ii)

We need to find the number of non-negative integer solutions to the system.

Using the method of stars and bars, we can write equation (i) as:

a + b + c + d + e + 5f = 25,

where f represents the number of bars or separators between the variables.

The number of non-negative integer solutions to this equation is:
(25+5-1) C (5-1) = 29 C 4 = 23751.

However, we need to ensure that equation (ii) is satisfied as well. We can do this by subtracting the number of solutions where a + b + c > 5 from the total number of solutions.

Let's consider the cases where a + b + c > 5.

Case 1: a + b + c = 6
In this case, we need to distribute the remaining 14 among a, b, c, d, e and f. This can be done using the stars and bars method:
(14+5-1) C (5-1) = 18 C 4 = 3060

Case 2: a + b + c = 7
In this case, we need to distribute the remaining 13 among a, b, c, d, e and f. This can be done using the stars and bars method:
(13+5-1) C (5-1) = 17 C 4 = 2380

Case 3: a + b + c = 8
In this case, we need to distribute the remaining 12 among a, b, c, d, e and f. This can be done using the stars and bars method:
(12+5-1) C (5-1) = 16 C 4 = 1820

Case 4: a + b + c = 9
In this case, we need to distribute the remaining 11 among a, b, c, d, e and f. This can be done using the stars and bars method:
(11+5-1) C (5-1) = 15 C 4 = 1365

Case 5: a + b + c = 10
In this case, we need to distribute the remaining 10 among a, b, c, d, e and f. This can be done using the stars and bars method:
(10+5-1) C (5-1) = 14 C 4 = 1001

Adding up the solutions in all the cases, we get:
3060 + 2380 + 1820 + 1365 + 1001 = 9626

Therefore, the required number of non-negative integer solutions is:
23751 - 9626 = 14125

Hence, the correct option is (B) 336.