A square is divided into 9 identical smaller squares. Six identical balls are to be placed in these smaller squares such that each of the three rows gets at least one ball (one ball in one square only). In how many different ways can this be done?

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Q: 81 (IAS/2005)
A square is divided into 9 identical smaller squares. Six identical balls are to be placed in these smaller squares such that each of the three rows gets at least one ball (one ball in one square only). In how many different ways can this be done?

question_subject: 

Maths

question_exam: 

IAS

stats: 

0,4,6,2,2,2,4

keywords: 

{'identical smaller squares': [0, 0, 1, 0], 'identical balls': [0, 0, 3, 0], 'smaller squares': [0, 0, 1, 0], 'square': [0, 0, 0, 1], 'rows': [0, 0, 1, 0], 'ball': [1, 3, 13, 12], 'many different ways': [0, 0, 8, 0]}

The question is about arranging 6 identical balls in 3 rows of a square that is divided into 9 smaller squares. Each row must contain at least one ball.

Option 1 - 27: This would be the number of ways if there are 3 balls and each ball is to be placed in a different row. But here we have 6 balls, so this option is incorrect.

Option 2 - 36: This would be the number of ways if there are 4 balls and each row should get at least one. However, we have 6 balls, so this option is incorrect.

Option 3 - 54: This would be correct if 6 balls were to be arranged across 9 squares without the condition that each row should get at least one ball. But here, each row must contain at least one ball, making this option incorrect.

Option 4 - 81: This is the correct answer. Since every row needs to have at least one ball, we can think of it as placing 3 balls in 3 rows, or 3^3 ways = 27 ways. Then for the remaining 3 balls, they can be placed in any of 9 squares, or 3^3 ways = 27 ways.

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