Question:

Step 1: Identify the given expression for calculation. The expression to be evaluated, as provided in the raw solution, is a product involving a binomial coefficient and a sum of terms. It is given as $^6C_3 [(3^7 - ^1C_1(2)^7 + ^2C_2]$. The solution then proceeds with numerical values for these terms. Step 2: Calculate the binomial coefficient. The first part of the expression is the binomial coefficient $^6C_3$. We calculate its value: $$^6C_3 = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(3 \times 2 \times 1)} = \frac{720}{6 \times 6} = \frac{720}{36} = 20$$ Step 3: Evaluate the sum within the square brackets. The raw solution directly provides the numerical values inside the square brackets as $[2187 - 384 + 3]$. We calculate this sum: $$2187 - 384 + 3 = 1803 + 3 = 1806$$ The original solution stated these terms as $3^7$, $^1C_1(2)^7$, and $^2C_2$, but then provided specific numerical values $2187$, $384$, and $3$ for these terms respectively in the calculation. Following the specific numerical calculation given in the raw solution, the sum is $1806$. Step 4: Multiply the results from Step 2 and Step 3. Now, we multiply the value of the binomial coefficient obtained in Step 2 with the sum obtained in Step 3: $$20 \times 1806 = 36120$$ Step 5: Conclude the final value. The final calculated value of the expression is $36120$. This calculation, labeled as "Option (B)" in the original solution, yields the value $36120$.