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

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$=1-\frac{D_{\left(15\right)}+{}^{15}C_1·D_{\left(14\right)}+{}^{15}C_2{D}_{\left(13\right)}}{15!}$

Where $D_n$ is de-arrangement and $D_n\approx \frac{n!}{e} \left\{\text{where }e\approx 2.73\right\}$

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

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

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

Note- This question was bonus as NTA as calculated the answer as $\frac{{}^{15}C_3\times 12!}{15!}=\frac{1}{6}$, they have not considered the repetation cases.