Permutations/Combinations (1 Viewer)

[goobi]

This question is from the Fitzpatrick textbook:

How many arrangements can be made of five letters chosen from the alphabet of 26 letters if no letter may be used twice in one word and no word may start with X nor with YZ in that order?

My solution:

Number of arrangements which exclude X and may include YZ at the beginning
= 25 x 24 x 23 x 22 x 21 = 6375600

Number of arrangements that start with YZ in this order
= 1 x 1 x 24 x 23 x 22 = 12144

Therefore, number of arrangements required
= 6375600 - 12144
= 6363456

However, the correct answer should be 7577856.

So can anyone please read over my solution and tell me what I did wrong?
Much appreciated!

 

[someth1ng]

If you look at the question, it says 5 letter combinations but it must NOT start with either X or YZ and this caused you to not answer the question properly.

Number of arrangements which exclude X and may include YZ at the beginning
= 25 x 24 x 23 x 22 x 21 = 6375600

This is not what the question asks, the question asks that it does not START with X but it can contain X

For permutations that start with X: X????
.'. n(combinations starting with X)=1x25x24x23x22

You do not multiply by 21 because that would be the 6th letter, there are four possible unknowns in the number you need to find - the first number when determining the permutations, the first is 1 and then you consider the other 4.

Number of arrangements that start with YZ in this order
= 1 x 1 x 24 x 23 x 22 = 12144

This bit is the part you did correctly and one of the things you needed to find the answer.

Therefore, number of arrangements required
= 6375600 - 12144
= 6363456

Now, that you've seen the first part, I don't understand what you are trying to find here. It's probably because you misunderstood the question causing you to answer it in a way that the question wasn't asking.

Solution
n(total combinations of 5 letters) - n(combinations starting with X) - n(combinations starting with YZ) = 26P5 - 1x25x24x23x22 - 1x1x24x23x22 = 7577856