Fifteen football players of a club-team are given 15 T-shirts with...

Given,

Fifteen football players of a club-team are given $15$ T-shirts with their names written on the backside

So, total sample space will be $15 !$

Now for at least $3$ players pick the correct T-shirt,

Required probability$= 1 - \dfrac{{D}_{\left(15\right)} + C1 15 \cdot {D}_{\left(14\right)} + C2 15 {D}_{\left(13\right)}}{15 !}$

Where ${D}_{n}$ is de-arrangement and ${D}_{n} \approx \dfrac{n !}{e}\left\{\text{where}e \approx 2 . 73\right\}$

Taking ${D}_{\left(15\right)}$ as $\dfrac{15\text{!}}{e}$, ${D}_{( 14 )}$ as $\dfrac{14 !}{e}$ and ${D}_{( 13 )}$ as $\dfrac{13 !}{e}$

We get , $1 - \left(\dfrac{\dfrac{15 !}{e} + 15 \cdot \dfrac{14 !}{e} + \dfrac{15 \times 14}{2} \times \dfrac{13 !}{e}}{15 !}\right)$

$= 1 - \left(\dfrac{1}{e} + \dfrac{1}{e} + \dfrac{1}{2e}\right) = 1 - \dfrac{5}{2e} \approx . 08$

Note- This question was bonus as NTA as calculated the answer as $\dfrac{C3 15 \times 12 !}{15 !} = \dfrac{1}{6}$, they have not considered the repetation cases.