Let $P(x)$ be the polynomial of degree at most $6$ which satisfies $P(k)=k!$ for $k=0,1,2,3,4,5,6$. Compute the value of $P(7)$.
To solve the problem, we will use the method of finite differences to find the value of \( P(7) \). The polynomial \( P(x) \) is of degree at most 6 and satisfies \( P(k) = k! \) for \( k = 0, 1, 2, 3, 4, 5, 6 \).

1. **List the values of \( P(k) \) for \( k = 0, 1, 2, 3, 4, 5, 6 \):**
   \[
   \begin{array}{c|c}
   k & P(k) = k! \\
   \hline
   0 & 1 \\
   1 & 1 \\
   2 & 2 \\
   3 & 6 \\
   4 & 24 \\
   5 & 120 \\
   6 & 720 \\
   \end{array}
   \]

2. **Compute the first-order finite differences:**
   \[
   \begin{array}{c|c}
   k & \Delta P(k) \\
   \hline
   0 & 1 - 1 = 0 \\
   1 & 2 - 1 = 1 \\
   2 & 6 - 2 = 4 \\
   3 & 24 - 6 = 18 \\
   4 & 120 - 24 = 96 \\
   5 & 720 - 120 = 600 \\
   \end{array}
   \]

3. **Compute the second-order finite differences:**
   \[
   \begin{array}{c|c}
   k & \Delta^2 P(k) \\
   \hline
   0 & 1 - 0 = 1 \\
   1 & 4 - 1 = 3 \\
   2 & 18 - 4 = 14 \\
   3 & 96 - 18 = 78 \\
   4 & 600 - 96 = 504 \\
   \end{array}
   \]

4. **Compute the third-order finite differences:**
   \[
   \begin{array}{c|c}
   k & \Delta^3 P(k) \\
   \hline
   0 & 3 - 1 = 2 \\
   1 & 14 - 3 = 11 \\
   2 & 78 - 14 = 64 \\
   3 & 504 - 78 = 426 \\
   \end{array}
   \]

5. **Compute the fourth-order finite differences:**
   \[
   \begin{array}{c|c}
   k & \Delta^4 P(k) \\
   \hline
   0 & 11 - 2 = 9 \\
   1 & 64 - 11 = 53 \\
   2 & 426 - 64 = 362 \\
   \end{array}
   \]

6. **Compute the fifth-order finite differences:**
   \[
   \begin{array}{c|c}
   k & \Delta^5 P(k) \\
   \hline
   0 & 53 - 9 = 44 \\
   1 & 362 - 53 = 309 \\
   \end{array}
   \]

7. **Compute the sixth-order finite differences:**
   \[
   \begin{array}{c|c}
   k & \Delta^6 P(k) \\
   \hline
   0 & 309 - 44 = 265 \\
   \end{array}
   \]

Since the sixth-order finite difference is constant (265), we can use it to find \( P(7) \).

8. **Backtrack to find \( P(7) \):**
   \[
   \Delta^5 P(6) = 309 \implies \Delta^5 P(7) = 309 + 265 = 574
   \]
   \[
   \Delta^4 P(6) = 362 \implies \Delta^4 P(7) = 362 + 574 = 936
   \]
   \[
   \Delta^3 P(6) = 426 \implies \Delta^3 P(7) = 426 + 936 = 1362
   \]
   \[
   \Delta^2 P(6) = 504 \implies \Delta^2 P(7) = 504 + 1362 = 1866
   \]
   \[
   \Delta^1 P(6) = 600 \implies \Delta^1 P(7) = 600 + 1866 = 2466
   \]
   \[
   P(6) = 720 \implies P(7) = 720 + 2466 = 3186
   \]

The final answer is \( \boxed{3186} \).