$2.$ To ensure a full line of outdoor clothing and accessories, the marketing department at Teddy Bower insists that they also sell waterproof hunting boots. Unfortunately, neither Teddy Bower nor Teddy Sports has expertise in manufacturing those kinds of boots. Therefore, Teddy Bower contacted several Chinese suppliers to request quotes. Due to competition, Teddy Bower knows that it cannot sell these boots for more than $$54.$ However, $$40$ per boot was the best quote from the suppliers. In addition, Teddy Bower anticipates excess inventory will need to be sold off at a $50$ percent discount at the end of the season. Given the $$54$ price, Teddy Bower$$s demand forecast is for $400$ boots, with a standard deviation of $300.$

a$)$ If Teddy Bower decides to include these boots in its assortment, how many boots should it order from its supplier?

b$)$ Suppose Teddy Bower orders $380$ boots. What would its expected profit be$?$