Inserting plus, times or nothing between digits

Question

Write the digits $1$ through $9$ in increasing order. Between each adjacent pair you may place a $+$ , a $*$ , or nothing (which glues the digits into a longer number), for instance $12+3*456+789$ or $123456789$ itself. How many different expressions can be formed?

Short answer

$6561$ . There are $8$ gaps between the nine digits and each gap independently gets one of three choices, so the count is $3^8=6561$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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