Question:

Step 1: Identify the formula for counting. The provided solution states that the number of ways is given by a specific formula. This formula is $2^n - 1$. $$ \text{Number of ways} = 2^n - 1 $$ Step 2: Understand the meaning of the formula. This formula, $2^n - 1$, is used to calculate the total number of non-empty subsets that can be formed from a set containing $n$ distinct objects. A non-empty subset means any subset excluding the empty set. Step 3: Determine the value of $n$ to match the correct answer. The problem asks for the number of roots, which is interpreted as the "number of ways" in the context of the provided solution. The correct answer provided is $7$ (Option 2). Therefore, we set the formula equal to $7$: $$ 2^n - 1 = 7 $$ Adding $1$ to both sides, we get: $$ 2^n = 8 $$ Since $8 = 2^3$, we can determine the value of $n$: $$ 2^n = 2^3 $$ $$ n = 3 $$ Thus, for the formula $2^n-1$ to yield $7$, the value of $n$ must be $3$. The final answer is $\boxed{7}$.