Question:

Step 1: Identify the expression to be evaluated. The problem requires evaluating an expression that uses binomial coefficients, powers, and the principle of inclusion-exclusion. The given expression is: $$ ^3C_x \times [3^7 - ^1C_1 2^7 + ^2C_2] $$ However, the calculation provided in the original solution uses specific numerical values for the binomial coefficients: $^3C_x$ is replaced by $10$, $^1C_1 2^7$ is replaced by $384$, and $^2C_2$ is replaced by $3$. The specific form of the expression being evaluated is: $$ 10 \times [3^7 - 384 + 3] $$ Step 2: Calculate the powers of the numbers involved. We need to calculate $3^7$. $$ 3^7 = 2187 $$ Step 3: Substitute the calculated power into the expression and perform the arithmetic within the brackets. Substitute $3^7 = 2187$ into the expression. $$ 10 \times [2187 - 384 + 3] $$ First, perform the operations inside the bracket: $$ 2187 - 384 + 3 = 1803 + 3 = 1806 $$ Step 4: Perform the final multiplication. Multiply the result from Step 3 by $10$: $$ 10 \times 1806 = 18060 $$ Step 5: Conclude with the final value and corresponding option. The evaluated value of the expression is $18060$. This corresponds to Option (B) in the original problem context. The final answer is $\boxed{18060}$.