Step-by-step Solution:

1. Total Letters and Grouping: \[ \text{There are 26 letters in the English alphabet.} \] \[ \text{We form a group of 7 letters: } a, \text{5 other letters, and } b. \] \[ \text{This leaves us with 19 more letters.} \] 2. Arrange the Objects: \[ \text{We have 20 objects in total: 1 group of 7 letters and 19 individual letters.} \] \[ \text{These 20 objects can be arranged among themselves in } 20! \text{ ways.} \] 3. Arrange the Group of 7 Letters: \[ \text{The group of 7 letters can start with either } a \text{ or } b. \] \[ \text{The number of ways to arrange the group is } 2 \times 5! \text{ (since the 5 letters in the middle can be arranged in } 5! \text{ ways).} \] 4. Select the 5 Letters for the Group: \[ \text{The 5 letters in the group can be selected from the remaining 24 letters (excluding } a \text{ and } b) \text{ in } \binom{24}{5} \text{ ways.} \] 5. Calculate the Total Number of Ways: \[ \text{The total number of ways is:} \] \[ \left(2 \times 5! \times \binom{24}{5}\right) \times 20! \] \[ \text{This can also be expressed as:} \] \[ 2 \times \binom{24}{5} \times 20! \] Therefore, the total number of ways is: \[ \boxed{2 \times \binom{24}{5} \times 20!} \]