1. **State the problem:** Find the number of distinct positive divisors of $30^4$ excluding 1 and $30^4$ itself. 2. **Prime factorization:** First, express 30 as a product of prime factors: $$30 = 2 \times 3 \times 5$$ 3. **Apply exponent:** Raise each prime factor to the 4th power: $$30^4 = (2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4$$ 4. **Number of divisors formula:** For a number expressed as $$n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$$ The total number of positive divisors is $$ (a_1 + 1)(a_2 + 1) \cdots (a_k + 1) $$ 5. **Calculate total divisors:** For $30^4 = 2^4 \times 3^4 \times 5^4$, the total number of divisors is $$ (4 + 1)(4 + 1)(4 + 1) = 5 \times 5 \times 5 = 125 $$ 6. **Exclude 1 and $30^4$:** Since the problem asks to exclude 1 and the number itself, subtract 2 from the total: $$125 - 2 = 123$$ 7. **Final answer:** The number of distinct positive divisors of $30^4$ excluding 1 and $30^4$ is $$\boxed{123}$$