A membrane for H$2/$CH$4$ separation was tested, and the permeate composition was $80\%$ H$2$ and the retentate composition was $42\%$ H$2$ when the feed was $50\%$ H$2.$ The absolute pressures of the feed and permeate were $100$ and $15$ lbf$/$in$2,$ respectively, and the permeate flow rate was $20\%$ of the feed flow rate. $($a$)$ Find the selectivity of the membrane. $($b$)$ Find the approximate permeate composition that can be achieved when the downstream side is vacuumed.