1. **Problem Statement:** We need to find the total number of distinct signals that can be created by arranging 3 pink, 3 white, and 2 black flags in a straight line. 2. **Formula Used:** When arranging $n$ objects where there are groups of identical objects, the number of distinct permutations is given by: $$\frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$ where $n$ is the total number of objects, and $n_1, n_2, \ldots, n_k$ are the counts of identical objects in each group. 3. **Applying the formula:** - Total flags: $3 + 3 + 2 = 8$ - Identical groups: 3 pink, 3 white, 2 black Number of distinct signals: $$\frac{8!}{3! \times 3! \times 2!}$$ 4. **Calculating factorials:** - $8! = 40320$ - $3! = 6$ - $2! = 2$ 5. **Substitute and simplify:** $$\frac{40320}{6 \times 6 \times 2} = \frac{40320}{72} = 560$$ 6. **Answer:** The total number of distinct signals that can be created is **560**.