Permutation Questions assuming that each letter can appear at most once in an arrangement Assuming that each letter can appear at most once in an arrangement, how many ways are there to arrange 10 letters taken from the alphabet a-z such that:
Exactly one of a and z appears in the arrangement?
z and a appear and are adjacent to each other in the arrangement?
For the first one, my current answer is 24!/15! as it would be 1 in the first spot which is locked up by a or z, then 24 letters left to go into 9 more spaces, decreasing by 1 each time due to no duplicates.
For the second, I thought it would be az, which could go into 5 spots so 5!, then 24!/16! in the remaining 8 spots, but that seems too large.
 A: The first question:

How many ways are there to arrange $10$ letters taken from the alphabet a - z such that exactly one of a and z appears in the arrangement?

You have not taken into account whether a or z has been selected or where it is located.
Choose whether a or z appears in the arrangement.  Choose which of the ten positions it occupies.  From the remaining $24$ letters (other than a and z), choose which will fill the leftmost open position.  From the remaining $23$ letters (other than a and z and the other letter that has already been selected), choose which will fill the leftmost open position.  Continue until all ten positions have been filled.

 $$2 \cdot 10 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 = 2 \cdot 10 \cdot \frac{24!}{15!}$$

The second question:

How many ways are there to arrange $10$ letters from the alphabet a - z such that a and z appear and are adjacent to each other in the arrangement?

You have not counted the number of places where the block containing a and z could appear correctly.  Also, you have to account for the order in which the letters a and z appear within the block.
Since the block containing a and z has length $2$, it must begin in one of the first nine positions.  Choose where the block begins.  Choose whether a or z appears first in the block.  Choose the letters for the remaining eight positions from the remaining $24$ letters, starting with the leftmost open position. 

  $$9 \cdot 2 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 = 9 \cdot 2 \cdot \frac{24!}{16!}$$

