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 example, for three items on the salad bar,
there are 7 different combinations of 1, 2, or 3 items on the
Super Bar.
If the three items are Bacon, Lettuce, and Cheese,
the seven combinations would include three combinations of
1 item
(1. Bacon, 2. Lettuce, and 3. Cheese),
three combinations of
2 items
(4. Bacon & Lettuce, 5. Bacon & Cheese, 6. Lettuce & Cheese),
and one combination of all three items
(7. Bacon, Lettuce, and Cheese).
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 combinations on the salad bar.
Here is a list of all the possible combinations:
| 15C1 | = | 15 (there are 15 ways to select one of the 15 items. |
| 15C2 | = | 105 (there are 105 ways to select 2 of the 15 items at a time) |
| 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 of those numbers.
Correctly solved by:
| 1. Kirstine Wynn |
St. Olaf's College, Northfield, Minnesota |
| 2. Richard K. Johnson | La Jolla, California |
| 3. George Gaither | Winchester, Virginia |
| 4. Tina Zahel | Winchester, Virginia |
| 5. Peggah Sadeghzadeh | Winchester, Virginia |
| 6. Kathleen Altemose | Winchester, Virginia |
| 7. Walt Arrison | Philadelphia, Pennsylvania |
| 8. Jeff Gaither | Winchester, Virginia |
| 9. Thai Tran | Ottawa, Canada |
| 10. Keith Mealy | Cincinnati, Ohio |
| 11. Daniel Wilberger | Winchester, Virginia |
| 12. David Powell | College of William & Mary, Williamsburg, Virginia |
| 13. Geoff Keith | Santa Monica, California |
| 14. Laneypoo11 | ---------- |
| 15. Sam Enumah | Columbus, Georgia |
| 16. Laura Snyder | Columbus, Georgia |