A seven-digit number of $a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}$ (all digits distinct) is formed random

Question

A seven-digit number of $a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}$ (all digits distinct) is formed randomly. The probability that number formed satisfy $a_{1} > a_{2} > a_{3} < a_{4} < a_{5} < a_{6} < a_{7}$ is

Accepted Answer

Sol. [Key Concept: $7$-digit number with distinct digits: count favorable arrangements.]7 digits are distinct and $a_{3}$ is the smallest∴ Number of favorable ways $=$$$\begin{matrix} P & \ = \frac{\left( \frac{10 	imes 9 	imes 8 	imes 6 	imes 5}{3 	imes 2 	imes 2} ight)}{9 	imes 9 	imes 8 	imes 7 	imes 6 	imes 5 	imes 4} \ & \ = \frac{5}{1512} \end{matrix}$$

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