The letters of the Latin Alphabet

Consider all \(26^{26}\) words of length 26 in the Latin alphabet. Define the weight of a word as \(\dfrac{1}{(k + 1)}\), where k is the number of letters not used in this word. Find the sum of the weights of all the words in Latin.

If the weight can be represented as \(a^b\), where \(a\) is in it's lowest form. That is if the answer is \(8^{66}\) , \(a\) is minimised when the answer is rewritten to be \(2^{198}\). Find \(ab\).

This is from the IMC 2015
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