Suppose that a "word" is any string of seven letters of the lower case english alphabet, with repeated letters allowed. how many "words" begin with r and end with t?

On the five remaining spaces for letters there are no restrictions (other than the letters must be lower case).

1 * 26 * 26 * 26 * 26 * 26 * 1

So all letters including the r and t can be reused. That means that you have 26^5 possibilities which is 11881376 possible words.