# Arrangements of all 26 letters [closed]

Consider all permutations of the twenty-six English letters that start with the letter $$z$$. In how many of these permutations the number of letters between $$z$$ and $$y$$ is less than those between $$y$$ and $$x$$?

I really couldn’t figure out how to do this one, can someone explain the solution

## closed as off-topic by user21820, Gibbs, YiFan, GNUSupporter 8964民主女神 地下教會, darij grinbergFeb 11 at 16:46

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Clearly $$x$$ must occur after $$y$$. So the options are
• $$y$$ occurs in place $$2$$ and $$x$$ occurs in place $$4$$ or later... $$23$$ possibilities;
• $$y$$ occurs in place $$3$$ and $$x$$ occurs in place $$6$$ or later... $$21$$ possibilities;
• $$y$$ occurs in place $$13$$ and $$x$$ occurs in place $$26$$... $$1$$ possibility.
Total, $$23+21+\cdots+1=144$$. Then arrange the other $$23$$ letters. Answer $$144\times23!\ .$$