International Mathematics Competition
for University Students
2020

IMC2020: Day 1, Problem 1

Problem 1. Let \(\displaystyle n\) be a positive integer. Compute the number of words \(\displaystyle w\) (finite sequences of letters) that satisfy all the following three properties:

   (1) \(\displaystyle w\) consists of \(\displaystyle n\) letters, all of them are from the alphabet \(\displaystyle \{\texttt{a},\texttt{b},\texttt{c},\texttt{d}\}\);

   (2) \(\displaystyle w\) contains an even number of letters \(\displaystyle \texttt{a}\);

   (3) \(\displaystyle w\) contains an even number of letters \(\displaystyle \texttt{b}\).

(For example, for \(\displaystyle n=2\) there are \(\displaystyle 6\) such words: \(\displaystyle \texttt{aa}\), \(\displaystyle \texttt{bb}\), \(\displaystyle \texttt{cc}\), \(\displaystyle \texttt{dd}\), \(\displaystyle \texttt{cd}\) and \(\displaystyle \texttt{dc}\).)

Armend Sh. Shabani, University of Prishtina

  Hint    Solution