Combinations

Question

A deli offers toppings for a sandwich: lettuce, tomato, cheese, onions, peppers, pickles, mayo and mustard. How many sandwiches can you make with AT LEAST TWO toppings

Answer 1

It's a bit of work, but entirely doable.If you used all 8 toppings, it wouldn't matter what order you used them, so there would be precisely one sandwich. If you used 7, there would be 8 different toppings you might individually leave out. It gets messier when you use six toppings, so we must now talk about the real method:

Combinations.

Remember the formula is: 

_nC_r  = n!      

          (n-r)!r!

That first answer took the form of:

₈C₈ = 1

The second is:

₈C_{7 = 8}

To get the rest, simply add the results of:

₈C₆

₈C₅

₈C₄

₈C₃

₈C₂

Remember that it said "at least two", so we have to stop our calculations there. Hint: this is less work than it seems. You will find that partway down the list, some familiar answers start reappearing.