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  4. Fifteen football players of a elub-team are given 15 T-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

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Q. Fifteen football players of a elub-team are given 15 T-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

JEE MainJEE Main 2023Probability - Part 2

Solution:

Required probability $=1-\frac{D_{(15)}+{ }^{15} C_1 \cdot D_{(14)}+{ }^{15} C_2 D_{(13)}}{15 !}$
Taking $D _{(15)}$ as $\frac{15 \text { ! }}{e}$
$D _{(14)}$ as $\frac{14 \text { ! }}{e}$
$D _{(13)}$ as $\frac{13 !}{e}$
We get, $1-\left(\frac{\frac{15 !}{e}+15 \cdot \frac{14 !}{e}+\frac{15 \times 14}{2} \times \frac{13 !}{e}}{15 !}\right)$
$=1-\left(\frac{1}{e}+\frac{1}{e}+\frac{1}{2 e}\right)=1-\frac{5}{2 e} \approx .08$