How many strings of eight uppercase English letters are there that start with X, if no letter can be repeated?

Question

mani_jha · Answer

_ _ _ _ _ _ _ _
Consider the eight spaces above to be filled by eight letters.
The first place can be filled in only 1 way(X)
The second place can be filled in any 25 ways(any of the remaining 25 letters)
The third place an be filled in any 24 ways(any of the remaining 24 letters)
..........
Similarly, the total  no of ways is 25X24X23..X1
=(25)!
Clear?

anonymous · Answer

Oh okay, but shouldn't it be 25*24*...*19 instead?

mani_jha · Answer

Oh yeah, I am sorry! You're correct.

anonymous · Answer

Haha thank you, I was thrown off by that fixed X as the first letter of the string..

mani_jha · Answer

If you know the formula for permutations, there's a more direct way of doing this:
The first letter is fixed,so now you've to choose 7 other letters from 25 letters(since X can't be used again)
So, no of ways will be 25P7
Both will give you the same answer.

anonymous · Answer

haha yeah, I was just lazy to type the whole fraction out..but thanks :D

anonymous · Answer

if the string should start with the letters BO (in this order), and if letters can be repeated, will it be 26^6?

mani_jha · Answer

Yes, exactly.