In how many ways can 12 distinct shirts be distributed among 3 people such that each person receives at least 1 shirt?
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.
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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