$4.$ A store is currently offering a $60\%$ discount on all items purchased. Your cashier is trying to convince you to open a store credit card and says to you, $"$ In addition to the $60\%$ discount you are receiving for purchasing these items on sale today, you will get an additional $20\%$ off for opening a credit card account. That means you are getting $80\%$ off!"

a$.$ What is the mistaken assumption here?

b$.$ Why is that assumption incorrect?

c$.$ If you did truly have $80\%$ discount, explain what should happen when you go to the counter to buy $$500$ worth of items? show calculation

d$.$ If you got your $60\%$ discount and opened the card for an additional $20\%$ what is the actual $\%$ discount you would receive? show calculation

e$.$ Is it better to apply the $60\%$ discount first or the $20\%$ discount first? show calculation