Forming Integers with Specific Digits

We need to form 4-digit numbers using $\{0, 1, 2, 3, 4\}$ that are $\le 4000$.

For 4-digit numbers $< 4000$:

• Thousands digit can be 1, 2, or 3 (3 options).
• Hundreds, tens, and units digits can be any of 5 digits (5 options each). Total $= 3 \times 5 \times 5 \times 5 = 375$.

In addition, $4000$ itself is formed by digits $4, 0, 0, 0$, which adds $1$ valid number.

Total numbers $= 375 + 1 = 376$.