Three positive real numbers $a, b, c$ satisfy $a^b = 343, b^c = 10, a^c = 7.$ Find $b^b.$
$$
\mathrm a. ~ 1000\qquad \mathrm b.~900\qquad \mathrm c. ~1200 \qquad \mathrm d. ~4000 \qquad \mathrm e. ~100
$$
1. Given the equations:
   \[
   a^b = 343, \quad b^c = 10, \quad a^c = 7
   \]
   We start by expressing \(a\) in terms of \(b\) and \(c\).

2. From \(a^b = 343\), we can write:
   \[
   a = 343^{1/b} = (7^3)^{1/b} = 7^{3/b}
   \]

3. From \(a^c = 7\), we can write:
   \[
   a = 7^{1/c}
   \]

4. Equating the two expressions for \(a\):
   \[
   7^{3/b} = 7^{1/c}
   \]

5. Since the bases are the same, we can equate the exponents:
   \[
   \frac{3}{b} = \frac{1}{c}
   \]

6. Solving for \(b\) in terms of \(c\):
   \[
   b = 3c
   \]

7. Substitute \(b = 3c\) into the equation \(b^c = 10\):
   \[
   (3c)^c = 10
   \]

8. We need to find \(b^b\). Using \(b = 3c\):
   \[
   b^b = (3c)^{3c}
   \]

9. From the equation \(b^c = 10\), we cube both sides to find \(b^b\):
   \[
   b^b = (b^c)^3 = 10^3 = 1000
   \]

Conclusion:
\[
\boxed{1000}
\]