How many combinations am I truly offering? I am currently running a promotion for my company and we would like to truly explain how many combinations exist for the product.  Here is how the steps break down:


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*Step $1:$ Customer orders Rice or Salad version 

*Step $2:$ Customer orders a protein: Chicken, Taco Beef, Steak or Veggie

*Step $3:$ Customer orders beans: black beans, pinto beans, refried beans (customer may order no beans, 1 bean, 2 beans, all bean options)

*Step $4:$ Add any or all of these toppings: Shredded Cheese, Pico, Romaine Lettuce Mix, Grilled Onions/Peppers, Fiesta Corn Salsa, Chipotle Citrus Vinaigrette, or Creamy Jalapeno Ranch. 

*Step $5:$ Add any or all of these toppings: Crumbled bacon, Guacamole, Sour Cream, Queso.

 A: Hint:  For each of the first two steps, the customer chooses one of a list, so you multiply the number of options at each step.  For the last three steps, you can make each item into a separate step and the choice is yes or no.  How many options is that?
A: Below, the $\binom{n}{k}$ notation refers to choose, which tells you the number of ways you can choose $k$ items from a collection of $n$. 
Step 1: 2 choices. 
Step 2: 4 choices. 
Step 3: It's the number of ways to choose $0$ items from $3$, plus the number of ways to choose $1$ item from $3$, etc.... $\binom{3}{0} + \binom{3}{1} +\binom{3}{2} +\binom{3}{3} = 8$ choices 
Step 4: Similar to step 3, there are 7 items, you can choose between 0 and 7 of them, so there are $\sum_{k=0}^{7} \binom{7}{k} = 2^7 = 128$ choices. 
Step 5: There are 4 items and you can choose between 0 and 4 of them, so there are 
$\sum_{k=0}^{4} \binom{4}{k} = 2^4 = 16$ choices
For the total number of combos you multiply these numbers together, so there are $2\cdot 4 \cdot 8 \cdot 128 \cdot 16 = 2^{17} = 131072$ total combos.
