(Solved): DISCRETE MATH PLEASE FOLLOW THE INSTRUCTIONS AND PROVIDE STEP BY STEP FOR EACH SUB-QUESTION Call ...

DISCRETE MATH

PLEASE FOLLOW THE INSTRUCTIONS AND PROVIDE STEP BY STEP FOR EACH SUB-QUESTION

Call a string of letters "legal" if it can be produced by concatenating (running together) copies of the following strings: 'v', 'ww', 'xxx' 'yyy' and ' \( z z z \) '. For each integer \( n \geq 1 \), let \( t_{n} \) be the number of legal strings with \( n \) letters. For example, \( t_{1}=1 \) (' \( v \) ' is the only the legal string). \( t_{2}= \) \( t_{n}=a t_{n-1}+b t_{n-2}+c t_{n-3} \) for each integer \( n \geq 4 \) where \( a=\quad b=\quad \), and \( c= \) For each integer \( n \geq 1 \), let \( p_{n} \) be the number of legal strings with \( n \) letters that also read the same right to left as they do left to right (like ' \( x x x v x x x^{\prime} \), for example). Which of the following expressions is equal to \( p_{101} \) ? \[ \begin{array}{l} t_{50}+3 t_{49} \\ t_{50}+3 t_{48} \\ p_{100}+p_{99} \\ t_{50}+t_{49} \\ p_{50}+p_{49} \\ t_{100}+t_{99} \\ p_{50}+3 p_{49} \\ t_{50}+t_{48} \end{array} \]