Determine the operations for A, B, and C.

We are given two equations:

1. $13 \text{ A } 14 \text{ B } 15 \text{ C } 20 = 243$
2. $15 \text{ A } 10 \text{ B } 8 \text{ C } 9 = 104$

Let's test common arithmetic operations. If we assume A is addition (+), B is multiplication (×), and C is addition (+), let's check if these operations satisfy both equations.

For equation 1:

$$13 + 14 \times 15 + 20$$

Following the order of operations (BODMAS/PEMDAS), multiplication is performed first:

$$13 + (14 \times 15) + 20$$ $$13 + 210 + 20$$ $$223 + 20 = 243$$

This matches the given result.

For equation 2:

$$15 + 10 \times 8 + 9$$

Following the order of operations:

$$15 + (10 \times 8) + 9$$ $$15 + 80 + 9$$ $$95 + 9 = 104$$

This also matches the given result.

Thus, the operations are:
A = + (addition)
B = × (multiplication)
C = + (addition)

The expression to evaluate is:

$$45 \text{ A } 10 \text{ B } 18 \text{ C } 50$$

Substitute the operations:

$$45 + 10 \times 18 + 50$$

Following the order of operations, perform multiplication first:

$$10 \times 18 = 180$$

Now substitute this value back into the expression:

$$45 + 180 + 50$$

Perform the additions from left to right:

$$45 + 180 = 225$$ $$225 + 50 = 275$$

The value of the expression is 275.

Comparing this with the given options:

1. 265
2. 280
3. 145
4. 275

The calculated value matches option 4.

The final answer is $\text{275}$.