Answer: 277 ways
Step-by-step explanation:
Let’s start bycreating 10-unit pieces using the 4-unit piece.
The arrangements are:
1). 4-3-3 (3 permutations)
2). 4-3-2-1 = 4! = 24 permutations.
3). 4-3-1-1-1 (5*[4!/(3!1!)]
= 5*4
= 20permutations
4). 4-2-2-2 (4 permutations)
5). 4-2-2-1-1 (5 *[4!/(2!2!)]
= 5*6
= 30 permutations
6). 4-2-1-1-1-1(6*[5!/(4!1!)]
= 6*5
= 30 permutations
Let’s-consider the arrangements using one or more3-unit pieces and no 4-unit piece:
7). 3-3-2-2 (4!/(2!2!)
8). 3-3-2-1-1 (5*(4!/(2!2!)
= 5*6
= 30 permutations.
9). 3-3-1-1-1-1 (6!/(4!2!) = 0
10). 3-2-2-2-1 (5*4!/(3!1!)
= 5*4
= 20permutations
11). 3-2-2-1-1-1 (6*5!/(3!2!)
= 6*10
= 60 permutations
Finally we would look at the arrangements using only 1-and 2-unit pieces:
12). 2-2-2-1-1-1-1 (7!/(4!3!)
= 35 permutations
add them all up:
(3 + 24 + 20 + 4) + (30 + 30 + 6 + 30) + (15 + 20 + 60 + 35)
=51 + 96 + 130
= 277ways
. Filling in accordingly, 
. This simplifies to 
and 36=6x. Therefore, x = 6 so the bases are 6 and 12.