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A software company surveyed 200 of its customers, every one of whom uses at least one of three tools: the editor, the debugger, and the profiler. Of the 200, exactly 150 use the editor, 130 use the debugger, and 110 use the profiler. What is the greatest possible number of these customers who use all three tools?
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Correct answer
A
Count total tool-memberships: 150 + 130 + 110 = 390 memberships among 200 customers, an excess of 190 over one membership each. Since every customer uses at least one tool, each customer falls in exactly 1, 2, or 3 tools. If e2 customers use exactly two tools and e3 use exactly three, then e2 + 2(e3) = 190. To maximize e3, push e2 to 0, giving 2(e3) = 190, so e3 = 95.
Another view uses deficits: 50 customers lack the editor, 70 lack the debugger, and 90 lack the profiler, for 210 missing memberships. A customer who uses all three tools has no deficits, so the other 200 − x customers must absorb all 210, and each of them can be missing at most two tools. Requiring 2(200 − x) ≥ 210 gives 200 − x ≥ 105, so x ≤ 95.
And 95 is achievable: 95 customers use all three tools, 55 use only the editor, 35 use only the debugger, and 15 use only the profiler. Then editor = 150, debugger = 130, profiler = 110, and the total is 200. Trying x = 96 fails, since the remaining 104 customers could absorb only 2 × 104 = 208 deficits, short of 210. So 95 is the greatest possible value. So the answer is A.
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