An 8 -character password is to be formed using a pool of 6 distinct uppercase letters and 5 distinct digits. The password must contain at least 4 letters and at least 3 digits. Furthermore, a specific letter and a specific digit must be included in the password, and they must be placed adjacent to each other. If the total number of valid passwords that can be formed is k 7! , then the value of k is

Step-by-step solution

First, we select the characters for the password. We need to choose 8 characters including at least 4 letters and at least 3 digits. Since 1 specific letter and 1 specific digit must be included, we need to select 6 more characters from the remaining 5 letters and 4 digits. To satisfy the overall condition, the remaining 6 characters must include at least 3 letters and at least 2 digits. Case 1: 4 letters and 2 digits Number of ways = ⁵C₄ ⁴C₂ = 5 6 = 30 Case 2: 3 letters and 3 digits Number of ways = ⁵C₃ ⁴C₃ = 10 4