How many 7 letter words exist that start with "C." Out of the last 6 letters 2 repeat. The "C" does not appear again and the 2 letters that repeat do not appear again and order is not important for the last 6 letters.
7 letter words have to be created starting with the letter C. The subsequent 6 letters have 2 pairs of the same letter. These do not occur in the word again. Also, the letter C is present only as the first letter. As the order of the 6 letters after C is not relevant it is combinations that is being dealt with here, not permutations.
The word can be divided into the following, a C in the beginning, 2 blocks consisting of 2 letters, and 2 blocks of single letters.
The number of letters from which to choose the first letter is 1, the first block of 2 letters has 25 letters to choose from, the second block of 2 letters has 24 letters to choose from, the other blocks have 23 and 22 letters to choose from. The total number of combinations is 25*24*23*22 = 303600
There are a total of 303600 letters with the given characteristics.