To find the number of ways to select 8 subjects from a total of 12 subjects, we can use the concept of combinations in combinatorial mathematics. The formula for combinations is given by: \[ C(n, r) = \frac{n!}{r!(n - r)!} \] Where: - \( n \) is the total number of items (in this case, subjects), - \( r \) is the number of items to choose, - \( ! \) denotes factorial, which is the product of all positive integers up to that number. ### Step-by-Step Explanation 1. **Identify the values of \( n \) and \( r \)**: - Here, \( n = 12 \) (the total number of subjects) and \( r = 8 \) (the number of subjects we want to select). 2. **Apply the combinations formula**: - We need to calculate \( C(12, 8) \): \[ C(12, 8) = \frac{12!}{8!(12 - 8)!} = \frac{12!}{8! \cdot 4!} \] 3. **Calculate the factorials**: - First, we can simplify \( 12! \) as follows: \[ 12! = 12 \times 11 \times 10 \times 9 \times 8! \] - This allows us to cancel \( 8! \) in the numerator and denominator: \[ C(12, 8) = \frac{12 \times 11 \times 10 \times 9 \times 8!}{8! \cdot 4!} = \frac{12 \times 11 \times 10 \times 9}{4!} \] 4. **Calculate \( 4! \)**: - \( 4! = 4 \times 3 \times 2 \times 1 = 24 \) 5. **Substitute back into the equation**: \[ C(12, 8) = \frac{12 \times 11 \times 10 \times 9}{24} \] 6. **Perform the multiplication in the numerator**: - Calculate \( 12 \times 11 = 132 \) - Then \( 132 \times 10 = 1320 \) - Finally, \( 1320 \times 9 = 11880 \) 7. **Divide by \( 24 \)**: \[ C(12, 8) = \frac{11880}{24} = 495 \] ### Conclusion Thus, the number of ways to select 8 subjects from 12 subjects is **495**. Therefore, the correct option is **B**. ### Explanation of Other Options - **Option A (490)**: This is incorrect because it underestimates the number of combinations. The calculations show that the correct number is higher. - **Option C (496)**: This is also incorrect. It is very close to the correct answer but does not account for all combinations correctly. - **Option D (498)**: This option is incorrect as well, being higher than the actual number of combinations. ### Revision Summary - Use the combinations formula \( C(n, r) = \frac{n!}{r!(n - r)!} \) to find the number of ways to choose items. - Factorials can simplify calculations by canceling common terms. - Always perform multiplication and division carefully to avoid errors. - The correct answer for selecting 8 subjects from 12 is **495** (Option B).