Question

a stock can go up, go down, or stay unchanged. how many possibilities are there if you own 9 stocks?
there are \\(\square\\) possibilities with 9 stocks.

Explanation:

Step1: Analyze the number of choices for each stock

Each stock has 3 possible states (go up, go down, or stay unchanged).

Step2: Use the multiplication principle

For \(n = 9\) stocks, the total number of possibilities is \(3\times3\times\cdots\times3\) (9 times). According to the rule of exponents, \(a^n=\underbrace{a\times a\times\cdots\times a}_{n\text{ times}}\). Here \(a = 3\) and \(n=9\), so the number of possibilities is \(3^9\).
Calculate \(3^9=3\times3\times3\times3\times3\times3\times3\times3\times3 = 19683\).

Answer:

\(19683\)