In 2008, Sonic Drive In ran an advertising campaign in which they boasted that they offered 168,894 combinations of drinks.   When asked how it computed the number, Sonic's PR department sent this response:  

"While the possible drink combinations at Sonic can be arrived at many different ways, the number 168,894 was generated by looking at how many different ways our 16 add-ins (Chocolate, Cherry, Strawberry, Diet Cherry, Pineapple, Lemon, Lime, Blue Coconut, Grape, Orange, Powerade, Cranberry, Apple Juice, Vanilla, Hi-C, Watermelon) could be applied to 12 base drinks on the menu.   While it's possible a customer could request all 16 add-ins into a drink thereby sky-rocketing the total number of drink combinations; we operated under the assumption that the most likely number of add-ins would be six or less to one drink."  

[Actually, the correct number should have been 178,716 combinations of drinks using their parameters].
Click here for an explanation of how I arrived at 178,716


Now in 2011, Sonic is advertising, "Bring your thirst to Sonic and enjoy half-price drinks and slushes from 2 to 4 p.m., every day.   More than 398,929 fountain drink and slush combinations to choose from."

Again, I believe that Sonic made an error in their computations.   Their website offers the following information about their drinks:

42 Basic drinks: choose from 10 Soft Drinks, 2 Iced Teas, 3 Flavored Teas, 3 Minute Maid Juices, 2 Milks, 6 Limeades, 4 Real Fruit Slushes, 7 Famous Slushes, 1 Ocean Water, 3 Fruit Smoothies, and 1 Water.

16 add-ins (Chocolate, Cherry, Strawberry, Diet Cherry, Pineapple, Lemon, Lime, Blue Coconut, Grape, Orange, Powerade, Cranberry, Apple Juice, Vanilla, Hi-C, Watermelon)

How many combinations are possible for 42 drinks and 16 add-ins if a customer only requests six or less add-ins per drink?

Solution to the Problem:

          There are 625,506 drink combinations using 42 basic drinks and 16 add-ins, but only using 6 or less add-ins per drink.

There are 8,008 different combinations of 16 add-ins, taken 6 at a time.
There are 4,368 different combinations of 16 add-ins, taken 5 at a time.
There are 1,820 different combinations of 16 add-ins, taken 4 at a time.
There are 560 different combinations of 16 add-ins, taken 3 at a time.
There are 120 different combinations of 16 add-ins, taken 2 at a time.
There are 16 different combinations of 16 add-ins, taken 1 at a time.
There is 1 combination of 16 add-ins, taken 0 at a time.

Adding these up, we get 14,893 different combinations for each drink.
Multiplying by 42 basic drinks gives us our answer of 625,506.

Correctly solved by:

No one.