If A = 555! And B = (278)555 then which one of the following relation is appropriate?Cannot be determinedA>BA<BA=B

To compare $A = 555!$ and $B = (278)^{555}$, we will analyze the two expressions mathematically and try to identify which relationship holds between them.

Step 1: Understanding Each Expression

1. Factorial Expression for A:

   • The factorial $555!$ represents the product of all positive integers from 1 to 555.
   • This value grows extremely rapidly as $n$ increases.

2. Exponential Expression for B:

   • The expression $B = (278)^{555}$ is an exponential expression where 278 is raised to the power of 555.

Step 2: Logarithmic Comparison

To compare the growth of both $A$ and $B$ more easily, we can use logarithms.

1. Taking Logarithms:

   • We compute $\log A$ and $\log B$ to compare their orders of magnitude.
   • From Stirling's approximation, we know: $n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n$
   • Thus, we can approximate: $\log(555!) \approx \frac{1}{2} \log(2 \pi \cdot 555) + 555 \log\left(\frac{555}{e}\right)$

2. Calculating $\log B$: $\log B = 555 \log(278)$

Step 3: Calculating and Comparing

Now we will compute both logarithmic values explicitly (or use approximations if needed):

1. Calculate $\log A$: $\log(555!) \text{ will give a very large value.}$

2. Calculate $\log B$: $\log B = 555 \log(278)$

For practical calculations, you can use numerical approximations:

• Using $\log_{10}(278) \approx 2.444$, we find: $\log B \approx 555 \times 2.444 \approx 1356.42$

3. Approximation for $\log A$:
   • A rough approximation, using Stirling's formula yields that $\log(555!)$ is significantly larger than $1356.42$ due to the factorial growth.

Step 4: Conclusion

Given that $\log(555!)$ is much greater than $555 \log(278)$, we conclude that: $A > B$

Final Answer

Thus, the appropriate relation between $A$ and $B$ is: A > B.