Question

1. At Wendy’s there are 11 options for burger toppings. Assuming you want a single burger and cannot do double toppings, how many different burgers can you order that have
   a. 0 toppings
   b.1 topping
   c. 2 toppings
   d. 6 toppings
   e. 8 toppings
   f. 9 toppings
   g. 10 toppings
   h. 11 toppings
   

Answer 1

Number of burgers each with k toppings out of 11 are  ¹¹C_k

a.

Number of burgers each with 0 toppings = ¹¹C₀   = 11! / ((11 - 0)! * 0!) = 1

b.

Number of burgers each with 1 topping1 = ¹¹C₁   = 11! / ((11 - 1)! * 1!) = 11! / 10! = 11

c.

Number of burgers each with 2 toppings = ¹¹C₂   = 11! / ((11 - 2)! * 2!) = (11 * 10) / 2 = 55

d.

Number of burgers each with 6 toppings = ¹¹C₆   = 11! / ((11 - 6)! * 6!) = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1) = 462

e.

Number of burgers each with 8 toppings = ¹¹C₈   = 11! / ((11 - 8)! * 8!) = (11 * 10 * 9) / (3 * 2 * 1) = 165

f.

Number of burgers each with 9 toppings = ¹¹C₉   = 11! / ((11 - 9)! * 9!) = (11 * 10) / ( 2 * 1) = 55

g.

Number of burgers each with 10 toppings = ¹¹C₀   = 11! / ((11 - 10)! * 10!) = 11 / 1 = 11

h.

Number of burgers each with 11 toppings = ¹¹C₁₁   = 11! / ((11 - 11)! * 11!) = 1