IMC 2015 Problem 8

Consider all $26^{26}$ words of length 26 in the Latin alphabet. Define the $\emph{weight}$ of a word as $1/(k+1)$ , where $k$ is the number of letters not used in this word. Prove that the sum of the weights of all words is $3^{75}$ .

Proposed by Fedor Petrov, St. Petersburg State University