Distributing objects in a grid structure subject to row constraints
Question
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?
Options
18
27
54
81
Explanation
The 9 cups sit in a 3 × 3 grid. We have 6 Coffee (C) cups and 3 Tea (T) cups. We require every single row to contain at least one C.
Instead of counting complex valid layouts, use complementary counting: Total unrestricted ways - Invalid ways. Total unrestricted ways to place 3 identical T cups in 9 slots = 9{3} = 9 × 8 × 7/3 × 2 × 1 = 84 ways.
An invalid arrangement is one where a row completely fails the rule, meaning a row contains ZERO coffees. A row with zero coffees must be filled entirely by the 3 Tea cups. Because there are exactly 3 Tea cups, they can only overwhelm a single row. Since there are 3 rows in the grid, there are exactly 3 ways to fill an entire row with Tea.
Valid layouts = Total - Invalid = 84 - 3 = 81.
Answer: (d).
Question details
Year
2022
Paper
CSAT
Question
Q77
Section
Numerical Ability
Sub-topic
Permutations & Combinations
Type
Factual single
Difficulty
Hard
Source hint
9 cups on 3x3 table, 6 coffee, 3 tea
Same sub-topic — other years
Permutations & Combinations has appeared in multiple CSAT papers:
See all questions on Permutations & Combinations
Browse every tagged question across all years