Answer :
To solve this problem, we need to determine which option matches a specific mathematical expression. The expression [tex]\(\binom{4}{5}^6\)[/tex] is presented in the problem.
However, [tex]\(\binom{4}{5}\)[/tex] refers to a combination function where you choose 5 items from a set of 4, which isn't possible. Therefore, [tex]\(\binom{4}{5}\)[/tex] is generally considered to be 0. Raising 0 to the 6th power will also result in 0. But since that doesn't directly help us match with any of the given options, let's check each option based on their numerical comparisons:
1. Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]:
- This is calculated as [tex]\((4^6)/(5^6)\)[/tex].
- The result of this calculation is approximately 0.262144.
2. Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]:
- This involves multiplying 6 by [tex]\(\frac{4}{5}\)[/tex].
- This calculation gives a value of approximately 4.8.
3. Option C: [tex]\(\frac{4^6}{5}\)[/tex]:
- This is [tex]\((4^6)/5\)[/tex].
- The result of this calculation is approximately 819.2.
4. Option D: [tex]\(\frac{24}{30}\)[/tex]:
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex].
- This fraction simplifies to 0.8.
Now, comparing these numerical results, Option A ([tex]\(\frac{4^6}{5^6}\)[/tex]) results in approximately 0.262144, which is the correct choice. This matches the comparative analysis we're performing, and thus Option A is the answer.
However, [tex]\(\binom{4}{5}\)[/tex] refers to a combination function where you choose 5 items from a set of 4, which isn't possible. Therefore, [tex]\(\binom{4}{5}\)[/tex] is generally considered to be 0. Raising 0 to the 6th power will also result in 0. But since that doesn't directly help us match with any of the given options, let's check each option based on their numerical comparisons:
1. Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]:
- This is calculated as [tex]\((4^6)/(5^6)\)[/tex].
- The result of this calculation is approximately 0.262144.
2. Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]:
- This involves multiplying 6 by [tex]\(\frac{4}{5}\)[/tex].
- This calculation gives a value of approximately 4.8.
3. Option C: [tex]\(\frac{4^6}{5}\)[/tex]:
- This is [tex]\((4^6)/5\)[/tex].
- The result of this calculation is approximately 819.2.
4. Option D: [tex]\(\frac{24}{30}\)[/tex]:
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex].
- This fraction simplifies to 0.8.
Now, comparing these numerical results, Option A ([tex]\(\frac{4^6}{5^6}\)[/tex]) results in approximately 0.262144, which is the correct choice. This matches the comparative analysis we're performing, and thus Option A is the answer.
