1. Interpretation

Each random assignment of 15 distinct T-shirts to 15 distinct players is a permutation of 15 objects.

We need the probability that at least 3 players get their own T-shirt, i.e. the number of fixed points in a random permutation is at least 3.

2. Useful fact about fixed points

For a random permutation of $n$ objects, the number of fixed points is approximately Poisson with mean $1$ when $n$ is reasonably large. Here $n=15$, and this approximation is very accurate.

So, $P(X=r) \approx e^{-1}\frac{1}{r!},$ where $X$ = number of players who pick the correct T-shirt.

We want $P(X\ge 3)=1-P(X=0)-P(X=1)-P(X=2).$

Using the Poisson$(1)$ model, $P(X=0)\approx e^{-1}, \quad P(X=1)\approx e^{-1}, \quad P(X=2)\approx \frac{e^{-1}}{2}.$

Hence $P(X\ge 3)\approx 1-e^{-1}\left(1+1+\frac12\right).$

Now, $e^{-1}\approx 0.3679,$ so $P(X\ge 3)\approx 1-0.3679\times 2.5=1-0.91975=0.08025.$

Thus, $P(X\ge 3)\approx 0.08.$

3. Match with the options

The closest and intended answer is:

$\boxed{0.08}$

So the correct option is D.

4. Verification with stored answer

Stored correct answer: D

Our derived answer: D

They agree.