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The number $15$ is expressed in the form

$15 = \frac{a _{1} ! a _{2} ! \dots a _{m} !}{b _{1} ! b _{2} ! \dots b _{n} !},$

where $a_{1} \geq a_{2} \geq \dots \geq a_{m}$ and $b_{1} \geq b_{2} \geq \dots \geq b_{n}$ are positive integers and $a_{1} + b_{1}$ is as small as possible. What is $∣ a_{1} - b_{1} ∣$ ?