1. **Problem statement:** We have an alphabet $\Omega = \{X, Y, Z, T, 0, 1, 2, 3, 4, 5, 6, 7\}$ with 12 characters, where 4 are letters ($X,Y,Z,T$) and 8 are digits ($0$ to $7$). We want to find $a_2$, the number of length-2 strings from $\Omega$ with no two consecutive numeric characters. 2. **Known values:** - $a_0 = 1$ (empty string) - $a_1 = 12$ (any single character from $\Omega$) 3. **Goal:** Calculate $a_2$ using a combinatorial argument. 4. **Step 1: Total possible pairs without restriction:** Each position can be any of the 12 characters, so total pairs = $12 \times 12 = 144$. 5. **Step 2: Count pairs with two consecutive numeric characters:** - Number of numeric characters = 8 - Number of pairs with two digits = $8 \times 8 = 64$ 6. **Step 3: Calculate $a_2$ by excluding invalid pairs:** $$a_2 = 12 \times 12 - 8 \times 8 = 144 - 64 = 80$$ 7. **Explanation:** - The $12 \times 12$ counts all possible pairs. - The $8 \times 8$ counts pairs where both characters are digits, which are not allowed. - Subtracting gives the number of valid pairs with no two consecutive digits. **Final answer:** $$a_2 = 80$$