The total number of words (with or without meaning) that can be formed out of the letters of the word "DISTRIBUTION" taken four at a time, is equal to ______.

"DISTRIBUTION" has 12 letters: D(1), I(3), S(1), T(2), R(1), B(1), U(1), O(1), N(1) — 9 distinct letters.

We count 4-letter words by cases based on repetition.

Case 1: All different. Choose 4 from 9 distinct letters: $$\binom{9}{4} \times 4! = 126 \times 24 = 3024$$.

Case 2: One pair + 2 different. Pairs available: I or T (2 choices). Choose 2 more from remaining 8: $$2 \times \binom{8}{2} \times \frac{4!}{2!} = 2 \times 28 \times 12 = 672$$.

Case 3: Two pairs. Choose 2 from $$\{I,T\}$$: $$\binom{2}{2} = 1$$. Arrangements: $$\frac{4!}{2!2!} = 6$$.

Case 4: Three same. Only I has 3 copies. Choose 1 more from 8: $$8 \times \frac{4!}{3!} = 32$$.

Total = $$3024 + 672 + 6 + 32 = \boxed{3734}$$.