Answer to September 16, 2002 Problem

Wendy's Salad Bar Problem

 
Several years ago, Wendy's advertisement for their Super Bar stated that they had "something for everyone. With over 26,335 possible combinations you can create just the dinner you want at the Super Bar."

What is the number of items on the Super Bar that would produce at least 26,335 combinations? You may include just 1 item, any combination of 2 items, any combination of 3 items, ... up to the total number of items on the Super Bar.

 

Solution to Problem:

The answer is 15.

For fourteen items on the salad bar,
there are 16,383 different combinations of 1, 2, 3, 4, ... up to 14 items on the Super Bar.

For fifteen items, there are 32,767 items on the salad bar.

Here is a list of all the possible combinations:
15C1 = 15
15C2 = 105
15C3 = 455
15C4 = 1,365
15C5 = 3,003
15C6 = 5,005
15C7 = 6,435
15C8 = 6,435
15C9 = 5,005
15C10 = 3,003
15C11 = 1,365
15C12 = 455
15C13 = 105
15C14 = 15
15C15 = 1
Sum = 32,767

Certainly, Wendy's has more than 15 items on their Salad Bar!!

Why did Wendy's use the number 26,335 in their advertisement?
I believe that possibly they were counting 229 items, and just summed the numbers from 1 to 229, which is exactly 26335. But this is erroneous since it only counts one combination of each number.


Correctly solved by:

1. Richard K. Johnson La Jolla, California
2. Renata Sommerville Austin, Texas
3.John Lybarger Calhoun Community College, Alabama
4. Sue Dutton West Malling, Kent, United Kingdom
5. Malcolm Harriss West Malling, Kent, United Kingdom
6. Joe Heintz Manchester, Tennessee


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