In a reading room of a library, there are 22 reading spots. Each reading spot…
2024
In a reading room of a library, there are 22 reading spots. Each reading spot consists of a round table with 9 chairs placed around it. Each table must have at least one reader, and the number of readers at each table must be unique. If there are 36 readers in total, how many reading spots will be left out without at least one reader?
Answer: C. 16 — Concept: The number of empty tables is fixed only if the seating is fixed. Here the only conditions are that every occupied table holds a different number of…
- A.
15
- B.
12
- C.
16
- D.
None of these
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Correct answer: C
Concept: The number of empty tables is fixed only if the seating is fixed. Here the only conditions are that every occupied table holds a different number of readers, each count is between 1 and 9, and the counts total 36. Several different seatings satisfy this, and they need not occupy the same number of tables, so the count of empty tables is a range, not one fixed value.
Application: The number of occupied tables k is any k for which 36 can be written as a sum of k distinct values from 1 to 9:
Fewest tables: use the largest counts. 9+8+7+6+5 = 35 < 36, so five tables cannot hold everyone; 9+8+7+6+5+1 = 36 uses 6 tables, leaving 22 - 6 = 16 empty.
Most tables: use the smallest counts. 1+2+3+4+5+6+7+8 = 36 uses 8 tables, leaving 22 - 8 = 14 empty. (Nine tables is impossible: the nine smallest distinct counts total 1+...+9 = 45 > 36.)
In between: 9+8+7+5+4+2+1 = 36 uses 7 tables, leaving 22 - 7 = 15 empty.
Cross-check: So a valid seating can occupy 6, 7 or 8 tables, leaving 16, 15 or 14 empty. The data fix neither the seating nor the number of occupied tables, so the number of empty spots is not uniquely determined. (12 empty is outright impossible: it needs 10 occupied tables, but at most 8 tables can carry all-different counts.)
Result: Because no single one of the offered numbers is forced by the given information, no specific numeric choice can be the answer, so the correct choice is none of these.