Explanation
1. Formulas
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Total ways to distribute 15 T-shirts: 15!
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Ways for exactly k players to pick the correct T-shirt: (k15)×D15−k
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Where Dn (derangement of n items) is: Dn=n!∑r=0nr!(−1)r
2. Solving for "At least 3 correct"
It is easier to calculate the complement: Total - (0 correct + 1 correct + 2 correct).
Case 0: Exactly 0 correct (Full Derangement)
Ways =(015)D15=15![0!1−1!1+2!1−3!1+⋯−15!1]
Case 1: Exactly 1 correct
Ways =(115)D14=15×14![0!1−1!1+2!1−3!1+⋯+14!1]
Ways =15![0!1−1!1+2!1−3!1+⋯+14!1]
Case 2: Exactly 2 correct
Ways =(215)D13=2!15×14×13![0!1−1!1+⋯−13!1]
Ways =2!15![0!1−1!1+2!1−⋯−13!1]
3. Probability Calculation
For large n, Dn≈en!.
P(at least 3)=1−[P(0)+P(1)+P(2)]
P(at least 3)≈1−[e1+e1+2!e1]
P(at least 3)≈1−e1(1+1+21)=1−e2.5
Since the options are simple fractions, we look at the specific terms of the expansion. For large n, the probability P(X=k)≈k!e1.
Summing the series for P(X≥3):
P(X≥3)=3!e1+4!e1+⋯≈e1(e−(1+1+21))=1−e2.5≈1−2.7182.5≈0.08
Checking the options:
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5/24≈0.208
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1/6≈0.166
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5/36≈0.138
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2/25=0.08
Correct Option: (4) 2/25
