1600 satellites were sent up by a country for several purposes. The purposes are classified as broadcasting (B), communication (C), surveillance (S), and others (O). A satellite can serve multiple purposes; however a satellite serving either B, or C, or S does not serve O. The following facts are known about the satellites: 1. The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2:1:1. 2. The number of satellites serving all three of B, C, and S is 100. 3. The number of satellites exclusively serving C is the same as the number of satellites exclusively serving S. This number is 30% of the number of satellites exclusively serving B. 4. The number of satellites serving O is the same as the number of satellites serving both C and S but not B. If the number of satellites serving at least two among B, C, and S is 1200, which of the following MUST be FALSE?

**Setup.** Use Venn regions: a = only B, b = only C, c = only S, d = B∩C only, e = B∩S only, f = C∩S only, g = B∩C∩S, and o = serves only "Others". **Apply the facts.** - Fact 2 → g = 100. - Fact 3 → b = c (call this m) and a = 10m/3. - |C| = |S| (from 2:1:1 ratio with C and S equal) forces d = e (call this D). - Fact 1's |B| = 2|C| simplifies to **f = 2m/3 − 50**. - Fact 4 → o = f. **Total = 1600** then collapses to **D = 800 − 10m/3**, giving |B| = 1700 − 10m/3, |C| = |S| = 850 − 5m/3. Non-negativity forces m to be a multiple of 3 with **75 ≤ m ≤ 240**. **This question.** "At least two among B/C/S" = d + e + f + g = 2D + f + 100 = 1200 ⇒ 2D + f = 1100 ⇒ m = **75** uniquely. So a = 250, D = 550, f = 0, o = 0, |B| = 1450, |C| = |S| = 725 — a single, fully determined configuration. - "C cannot be uniquely determined" — **FALSE**, since |C| = 725 is fixed. - "B is more than 1000" — true (1450). - "All 1600 satellites serve B or C or S" — true (o = 0). - "B exclusively is exactly 250" — true (a = 250). **Answer: A** — the only statement that *must* be false.