Placing operators among digits

MediumCombinatorics~4m

How many arithmetic expressions can be formed from the string $123456789$ (digits kept in order) by choosing, in each of the $8$ gaps between consecutive digits, to either concatenate the digits, insert a $+$ , or insert a $\times$ ? (For example, $1 + 23 \times 4 + \dots$ )

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There are $8$ gaps between the $9$ digits.
Each gap has $3$ independent choices (join, $+$ , $\times$ ): $3^8$ .

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Asked at: Jane Street, Citadel

Placing operators among digits

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