Fifteen football players of a club-team are given `15T` -shirts with their names written on the backside.If the players pick up the T-shirts randomly,then the probability that at least `3` players pick the correct `T` shirt is

Options

A. `(5)/(36)`
B. `(2)/(15)`
C. `(5)/(24)`
D. `(1)/(6)`

Answer

To solve the problem of finding the probability that at least 3 players pick the correct T-shirt, we can use the principle of complementary counting. We will first calculate the probability that fewer than 3 players pick the correct T-shirt, and then subtract this from 1.

### Step-by-step Solution:

1. **Define the Total Outcomes**:
The total number of ways the 15 players can pick their T-shirts is given by the total permutations of 15 items, which is \(15!\).

**Hint**: Remember that each player can pick any of the 15 shirts, leading to \(15!\) total arrangements.

2. **Calculate the Probability of Fewer than 3 Correct Picks**:
We need to find the cases where 0, 1, or 2 players pick the correct T-shirt.

- **Case 1: 0 Correct Picks**:
This is a derangement problem, where none of the players pick their correct T-shirt. The number of derangements \(D_n\) of \(n\) items can be calculated using the formula:
\[ D_n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!} \]
For \(n = 15\):
\[ D_{15} = 15! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \ldots + \frac{(-1)^{15}}{15!} \right) \]

- **Case 2: 1 Correct Pick**:
Choose 1 player to pick their correct T-shirt (there are \( \binom{15}{1} \) ways to choose the player). The remaining 14 players must pick incorrectly, which is \(D_{14}\):
\[ \text{Ways} = \binom{15}{1} \cdot D_{14} \]

- **Case 3: 2 Correct Picks**:
Choose 2 players to pick their correct T-shirts (there are \( \binom{15}{2} \) ways). The remaining 13 players must pick incorrectly, which is \(D_{13}\):
\[ \text{Ways} = \binom{15}{2} \cdot D_{13} \]

3. **Combine the Cases**:
The total number of ways for fewer than 3 correct picks is:
\[ \text{Total} = D_{15} + \binom{15}{1} D_{14} + \binom{15}{2} D_{13} \]

4. **Calculate the Probability**:
The probability that fewer than 3 players pick the correct T-shirt is:
\[ P(\text{fewer than 3}) = \frac{D_{15} + \binom{15}{1} D_{14} + \binom{15}{2} D_{13}}{15!} \]

5. **Find the Required Probability**:
The probability that at least 3 players pick the correct T-shirt is:
\[ P(\text{at least 3}) = 1 - P(\text{fewer than 3}) \]

6. **Final Calculation**:
Substitute the values and simplify to find the final answer. After performing the calculations, we find:
\[ P(\text{at least 3}) \approx 0.08 \]

### Final Answer:
The probability that at least 3 players pick the correct T-shirt is approximately **0.08**.