1. **Problem Statement:** Let $A$, $B$, and $C$ be distinct nonzero digits such that their product is $12$. We form all possible 3-digit numbers using $A$, $B$, and $C$ without repetition. Let $X$ be the sum of all these numbers. We want to analyze the statements: I. The least value of $X$ is less than 5000. II. The greatest value of $X$ is above 5000. 2. **Step 1: Find all possible triples $(A,B,C)$ of distinct nonzero digits with product 12.** Since $A,B,C$ are digits from 1 to 9 and distinct, find all sets where $A \times B \times C = 12$. Possible factorizations of 12 into three distinct digits: - $(1,2,6)$ since $1 \times 2 \times 6 = 12$ - $(1,3,4)$ since $1 \times 3 \times 4 = 12$ No other distinct digit triples multiply to 12. 3. **Step 2: Calculate $X$ for each triple.** The 3-digit numbers formed by $A,B,C$ without repetition are all permutations of these digits. There are $3! = 6$ permutations. Sum of all permutations for digits $A,B,C$: Each digit appears in each place (hundreds, tens, units) exactly 2 times (since 6 permutations and 3 digits). So, $$X = 2 \times 100 (A+B+C) + 2 \times 10 (A+B+C) + 2 \times 1 (A+B+C) = 2 (100 + 10 + 1)(A+B+C) = 2 \times 111 \times (A+B+C) = 222 (A+B+C)$$ 4. **Step 3: Calculate $X$ for each triple:** - For $(1,2,6)$: $A+B+C = 1+2+6 = 9$ $$X = 222 \times 9 = 1998$$ - For $(1,3,4)$: $A+B+C = 1+3+4 = 8$ $$X = 222 \times 8 = 1776$$ 5. **Step 4: Analyze the statements:** - Least value of $X$ is $1776$ (from $(1,3,4)$), which is less than 5000. So statement I is **true**. - Greatest value of $X$ is $1998$ (from $(1,2,6)$), which is less than 5000. So statement II is **false**. **Final answers:** - Statement I: True - Statement II: False