Rosie$$s Company has three products, P$1,$ P$2,$ and P$3.$ The maximum Rosie$$s can sell is $65,000$ units of P$1,24,000$ units of P$2,$ and $12,000$ units of P$3.$ Rosie$$s has limited production capacity of $108,000$ machine hours. Machine hours per unit are as follows: P$1,1$ hour; P$2,2$ hours; P$3,4$ hours. Contribution margin per unit is $$5$ for P$1,$ $$15$ for P$2,$ and $$25$ for P$3.$ What is the most profitable sales mix for Rosie$$s Company? $($Give step by step instructions$)$