Question 64
Find the number of words using $2014$ letters $a$ and $2014^2$ letters $b$ with the following properties:
i) Among the first $k$ letters of the word, number of $b$'s is not less than number of $a$'s for any $k$ satisfying the condition $i \le k \le 2014 \cdot 2015$;
ii) Among the last $k$ letters of the word, number of $b$'s is not less than number of $a$'s for any $k$ satisfying the condition $i \le k \le 2014 \cdot 2015$.
i) Among the first $k$ letters of the word, number of $b$'s is not less than number of $a$'s for any $k$ satisfying the condition $i \le k \le 2014 \cdot 2015$;
ii) Among the last $k$ letters of the word, number of $b$'s is not less than number of $a$'s for any $k$ satisfying the condition $i \le k \le 2014 \cdot 2015$.