A local coffee store announces an offer for $7$ days. Customers would receive one coupon for every two days they purchased coffee at the store. Let N be the number of days a customer purchases coffee and C be the number of coupons the customer earns over the $7$ days. Suppose a customer purchases coffee on any day with probability p independently of other days. 

Answer the following questions.

$($a$)$ Derive the joint PMF of N and C$.$

$($b$)$ Derive the average number of coupons a customer earns.

$($c$)$ Derive the average number of days a customer purchases coffee.