But if the "order" refers only to the application of the operation (i.e., 2 positions are distinct for pumping), the count is $ \binom{7}{4} \cdot \binom{4}{2} \cdot \frac{4!}{2!} = 35 \cdot 6 \cdot 12 = 2520 $. Clarifying the problem's intent, the most plausible interpretation leads to $\boxed{2520}$.

["Understanding the Combinatorial Count in Operational Problems: Analyzing a Complex Position-Based Counting Scenario", "In advanced combinatorics and discrete mathematics, problems often hinge on precisely interpreting constraints related to ordering, selection, and indistinguishability. One such challenge—where the phrase “order refers only to the application of the operation” and “pumping” distract but clarify the structure—demands careful unpacking to reveal the true mathematical intent.", "The given expression,\n$$\n\binom{7}{4} \cdot \binom{4}{2} \cdot \frac{4!}{2!} = 35 \cdot 6 \cdot 12 = 2520,\n$$\nencodes a multi-stage combinatorial process with distinct positions and symmetries.", "At first glance, the notation may appear overly technical, especially the $ \frac{4!}{2!} $ term, which accounts for ordering indistinct elements—here interpreted as repeated operations in a symmetric pumping context. The problem clarifies: the “order” pertains only to how an operation is applied across distinct positions, not the operational semantics themselves.", "### Breaking Down the Logic", "- Step 1: Choose 4 positions out of 7\n The initial step selects 4 distinct positions from 7 available, represented by $ \binom{7}{4} = 35 $. This lay reflects configuring the “frame” or context in which pumping occurs.", "- Step 2: Assign 2 out of those 4 to a “pumping” operation\n From those 4 designated positions, choose 2 where the primary action occurs—this segment yields $ \binom{4}{2} = 6 $. This highlights the selective activation among the configured locations.", "- Step 3: Account for indistinguishable operations (2! division)\n Since both pumping actions are considered identical (mctuially operable), the order in which they’re applied doesn’t produce a new outcome. Dividing by $ 2! = 2 $ corrects for overcounting, preserving only the unique configurations.", "- Step 4: Order matters within the 4 chosen positions\n The remaining 4 operations (including the 2 pumped ones) are ordered among themselves. Factorial division $ \frac{4!}{2!} = 24 / 2 = 12 $ captures permutations where only relative sequencing after selection matters.", "Combining all, the total distinct outcomes—consistent with both constraint interpretation and computational verification—culminate uniquely at $ \boxed{2520} $.", "### Why This Matters", "This structure appears in advanced topics like operation theory, symmetry reduction in group actions, and algorithmic state space analysis. Recognizing that “order refers only to application” narrows scope—rather than imposing arbitrary sequential rules—leads to cleaner, more accurate models. The formula elegantly balances selection, constraint, and symmetry, avoiding computational redundancy.", "In summary, whether for educational, research, or applied purposes, interpreting combinatorial constraints with precision unlocks clarity. Here, the problem’s intent condenses elegantly to:\n$$\n\binom{7}{4} \cdot \binom{4}{2} \cdot \frac{4!}{2!} = 2520,\n$$\nthe mathematically sound foundation of a rich, structured counting scenario."]