There are 9 cups placed on a table arranged in equal number of rows and columns out of which 6 cups contain coffee and 3 cups contain tea. In how many ways can they be arranged so that each row should contain at least one cup of coffee?
- (a) 18
- (b) 27
- (c) 54
- (d) 81 ✓ UPSC's answer
Why the answer is (d)
• 9 cups in equal rows and columns form a 3 × 3 grid. The arrangement is fixed by choosing which 3 of the 9 places hold tea: 9C3 = 84 ways.
• A row has no coffee only if all 3 of its cups are tea. Since there are exactly 3 tea cups, this can happen in only one row at a time — 3 arrangements (one for each row).
• Arrangements with at least one coffee cup in every row = 84 − 3 = 81, option (d).
Why the other options are wrong
- (a) 18
- The correct count is 9C3 − 3 = 84 − 3 = 81, not 18.
- (b) 27
- The correct count is 9C3 − 3 = 84 − 3 = 81, not 27.
- (c) 54
- The correct count is 9C3 − 3 = 84 − 3 = 81, not 54.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.