Discussion

Explanation:

Here shirts are similar, and people are distinct.

Total number of ways = 3¹² 

This includes those cases when 1 or 2 people don’t receive any shirt.

Case 1: 2 people don’t receive any thing i.e., 1 receives all the 12 shirts.

Number of ways of selecting 1 person out of 3 = ³C₂ = 3 ways.

Now number of ways of giving 12 shirts to this person = 1.

∴ Total number of ways in which only 1 person receives all the 3 shirts = 3 × 1 = 3 ways.

Case 2: 2 people receive at least 1 shirt, and 1 person doesn’t get any shirt.

Number of ways of selecting 1 person out of 3 = ³C₂ = 3 ways.

Now number of ways of giving 12 shirts to 2 other persons = 2¹² - 2.

[2¹² is total number of ways of distribution 12 shirts to 2 people. There will be 2 cases here where 1 person receive all 12 shirts.]

∴ Total number of ways in which only 2 persons receives all the 3 shirts = 3 × (2¹² - 2) ways.

∴ Total number of ways in which every person receives at least 1 shirt = 3¹² – 3 – 3(2¹² - 2) = 3¹² – 3 × 2¹² + 3 = 3¹² – 3069.

Hence, option (a).

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