# International Mathematics Competition for University Students 2020

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IMC 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

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