Question 1135813
<b>How many seven letter code words can be formed using a standard 26 letter alphabet if the first letter has to be a C and the rest of letters are different?</b>
<pr>
It's not exactly clear if the rest of the letters have to be different from just the 'C' or also different from one another (including the 'C').  I'll assume it's the latter, that all seven letters need to be different:
<pr>
1 * 25 * 24 * 23 * 22 * 21 * 20 = 127,512,000
<pr>
<b>And if the first two letters have to be a vowel and the rest of the letters are different?</b>
<pr>
Again, I'm assuming that "the rest of the letters are different" means all the letters need to be different from one another, and not only different from the first two letters.  However, when it says "the first two letters have to be a vowel" it does NOT say that the first two letters are not allowed to be identical vowels.  Taking all this into consideration:
<pr>
[(4/5) * 5 * 4 * 24 * 23 * 22 * 21 * 20] + [(1/5) * 5 * 1 * 25 * 24 * 23 * 22 * 21] = 87,983,280
<pr>
What is happening here is that, 4/5 of the time, the first two vowels will be different.  1/5 of the time, the first two vowels will be the same.