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View Full Version : Sonic Drink Promotion? (Probabilities and Statistics)

Gacker
08-21-2006, 12:38 PM
My wife is a middle school math teacher, and saw the promotion at Sonic that claims 168,894 soda drink combinations. She wants to use it as an exmple for her students, so she asked me how they arrive at that number. This challenge has casued the inner geek in me to emerge, so I emailed Sonic and asked for a few more details.

Their response was:
12 sodas, 16 potential mix-ins but no more than 6 mix-ins per soda. So, I sat down to run some numbers.

Assumption: I am going to have to assume for now that only 1 soda will be used per drink. They were not clear on this in their answer.

We should be able to agree that order is not important in mixing the drinks, so we are looking at combinations and not permutations. Therefore, a well known formula for counting combinations will be used:

nCr = n!/(r!(n-r)!)

The first calculation is to figure out how many combinations of mix-ins there are using Sonic's claim of 16 total mix-ins with no more than 6 per soda. I believe it reduces to the following:

Mix-in Combinations = 16C6 + 16C5 + 16C4 + 16C3 + 16C2 + 16C1
Mix-in Combs = 8008 + 4368 + 1820 + 560 + 120 + 16
Mix-in Combs = 14,892

Finally, multiplying the number of sodas times the potential mix-in combinations:

Total drinks = 12 sodas x 14,892 mix-in combinations + 12 sodas with no mix-ins
Total drinks = 178,704 + 12
Total drinks = 178,716

My number is 9,822 drinks higher than Sonic claims. Does anyone see an error in my calculations that would explain this discrepancy before I email Sonic back and question the validity of their number?

If my assumption is invalid, than the number goes even higher.

Jameson
08-21-2006, 04:37 PM
Hello Gacker!

I checked over your thought process and calculations and it all looks good! Either we're misinterpreting the setup or Sonic needs to higher people who have taken some math classes.

Good post.

Jameson

arbit
08-22-2006, 04:58 PM
of all the potential combinations ..some wud not be tasting gud and thts why those wud have to be eliminated ..of which no info is available