JEE MAIN - Mathematics (2022 - 27th June Morning Shift - No. 14)
Five numbers $${x_1},{x_2},{x_3},{x_4},{x_5}$$ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order $$({x_1} < {x_2} < {x_3} < {x_4} < {x_5})$$. The probability that $${x_2} = 7$$ and $${x_4} = 11$$ is :
$${1 \over {136}}$$
$${1 \over {72}}$$
$${1 \over {68}}$$
$${1 \over {34}}$$
Explanation
No. of ways to select and arrange $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4, \mathrm{x}_5$ from $1,2,3$.......18
$$ \begin{aligned} & \mathrm{n}(\mathrm{s})={ }^{18} \mathrm{C}_5 \\\\ & \begin{array}{lllll} x_1 & \underset{7}{\left(x_2\right)} & x_3 & \underset{11}{\left(x_4\right)} & x_5 \end{array} \\\\ & \mathrm{n}(\mathrm{E})={ }^6 \mathrm{C}_1 \times{ }^3 \mathrm{C}_1 \times{ }^7 \mathrm{C}_1 \\\\ & P(E)=\frac{6 \times 3 \times 7}{{ }^{18} C_5} \\\\ & \frac{1}{17 \times 4}=\frac{1}{68} \\\\ \end{aligned} $$
$$ \begin{aligned} & \mathrm{n}(\mathrm{s})={ }^{18} \mathrm{C}_5 \\\\ & \begin{array}{lllll} x_1 & \underset{7}{\left(x_2\right)} & x_3 & \underset{11}{\left(x_4\right)} & x_5 \end{array} \\\\ & \mathrm{n}(\mathrm{E})={ }^6 \mathrm{C}_1 \times{ }^3 \mathrm{C}_1 \times{ }^7 \mathrm{C}_1 \\\\ & P(E)=\frac{6 \times 3 \times 7}{{ }^{18} C_5} \\\\ & \frac{1}{17 \times 4}=\frac{1}{68} \\\\ \end{aligned} $$
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