Sonic drink combinations

[jtsm...@gmail.com]

I recently visited Sonic Drive In and read about their current
advertising campaign. They boast 168894 combinations of drinks. After
coming home and working for an hour or so at coming up with this
number, I sent an email to Sonic's PR department asking how they came
up with it. Well, this is the response I recieved:
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.
With this information, I still cannot reach the number given in the
advertising campaign. Can anyone else reach this figure, and if so,
can you please respond as to how. Much appreciation in advance.

-Taylor Smith

[mensa...@aol.com]

jtsm...@gmail.com wrote:
> I recently visited Sonic Drive In and read about their current
> advertising campaign. They boast 168894 combinations of drinks. After
> coming home and working for an hour or so at coming up with this
> number, I sent an email to Sonic's PR department asking how they came
> up with it. Well, this is the response I recieved:
> 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

c.append(gmpy.comb(16,6))

> or less

c.append(gmpy.comb(16,5))
c.append(gmpy.comb(16,4))
c.append(gmpy.comb(16,3))
c.append(gmpy.comb(16,2))
c.append(gmpy.comb(16,1))
c.append(gmpy.comb(16,0))
c
[mpz(8008), mpz(4368), mpz(1820), mpz(560), mpz(120), mpz(16), mpz(1)]

sum(c)
mpz(14893)

> to one drink.

sum(c)*12
mpz(178716)

> With this information, I still cannot reach the number given in the
> advertising campaign.

Now you know why the Liberal Arts grads go into advertising.

> Can anyone else reach this figure,

Not likely.

> and if so,
> can you please respond as to how.

Sorry, all I can do is give the correct answer.

[Jens Kruse Andersen]

mensanator wrote:

> jtsm...@gmail.com wrote:
> > 168,894 was generated by looking at how many
> > different ways our 16 add-ins

> > [...]

> > could be applied to 12 base drinks on the menu.

> mpz(178716)

>
> > With this information, I still cannot reach the number given in the

> > advertising campaign. Can anyone else reach this figure,
>
> Not likely.

PARI/GP agrees with mensanator:
(13:30) gp > sum(i=0,6,binomial(16,i)) * 12
%80 = 178716

12 does not divide 168894 = 2 * 3^2 * 11 * 853, so I see no interpretation
which might work.
http://www.sonicdrivein.com/pdfs/newsReleases/06_06_26_01.pdf both says
"168,894 to be exact" and "more than 168,894" on the same page.
Hopefully they are better at drinks than maths.

--
Jens Kruse Andersen

[jtsm...@gmail.com]

I'm trying to also use the basis of the method with different starting
numbers to reach this number as well. For instance, what if there are
18 base drinks (which does divide 168894) and then use 16 add-ins, or
perhaps 15. If you use this said combination, and look at

15 choose 6 = 5005
15 choose 5 = 3003
15 choose 4 = 1365
-----------------------------
Add those up, you get 9373, 9373*18= 168714
This answer is 180 off the one I'm looking for. Its as close as I can
get so far. What I was wondering, is if anyone can obtain 168894 by
fiddling with the numbers, in a somewhat logical manner.

-Taylor Smith

[Jens Kruse Andersen]

jtsmith2 wrote:
> I'm trying to also use the basis of the method with different starting
> numbers to reach this number as well.

> What I was wondering, is if anyone can obtain 168894 by
> fiddling with the numbers, in a somewhat logical manner.

A computer search says 17 add-ins with 2 to 5 used in 18 base drinks
would give 168912 combinations.
Forcing at least 2 add-ins seems odd, and not a single of the parameters
match the explanation you got.

I think they either miscalculated or have hidden assumptions, e.g. about
which add-ins can be combined with which base drinks, or that some
add-ins like Cherry and Diet Cherry cannot be combined.
Eliminating the case where the only two add-ins are Cherry and Diet Cherry
would give the "right" result in my example, but then allowing them both
if 1 to 3 more add-ins are used would not be logical.

--
Jens Kruse Andersen

[jtsm...@gmail.com]

Very nice. Ok, so at this point, the actual computations are getting
beyond the scope of my math knowledge. What if we removed the case
where Cherry and Diet Cherry were able to be added in the same drink,
but still able to be added to other drinks individually.

I'm honestly not sure how to go about doing those calculations, so I'm
hoping Jens or someone else might know. If that is taken into account,
I believe it might change Jens' outcomes so that one might land exactly
on the "right" number (ok, so probably not, but worth a shot).

I'm under the impression that it was a miscalculation on their part,
but I would like to search all corners on the slight chance that
they're correct.

Thank you all for you input and help.

-Taylor Smith

[Jens Kruse Andersen]

jtsmith2 wrote:
> Very nice. Ok, so at this point, the actual computations are getting
> beyond the scope of my math knowledge. What if we removed the case
> where Cherry and Diet Cherry were able to be added in the same drink,
> but still able to be added to other drinks individually.

r out of n add-ins can be combined in n choose r ways.
Subtracting all those where both Cherry and Diet Cherry are present gives:
(n choose r) - ((n-2) choose (r-2))

I already tried that and cannot hit the claimed answer.

> I'm under the impression that it was a miscalculation on their part,
> but I would like to search all corners on the slight chance that
> they're correct.

I don't think the claimed answer can be reached by simple interpretations
and modifications of the info you got.
It's possible the info is horribly inadequate. They offered a prize of
$168,894 so one would think they thought about the number,
but I don't want to do more idle speculation.

--
Jens Kruse Andersen

[jtsm...@gmail.com]

I figured I would follow up with the correct combinations from a guy at
Sonic who emailed me back this afternoon.
-------------
Taylor,

Good afternoon and thanks for your comments. Actually,
there are two pieces of info that got left out.

First, we have 8 "base drinks" and 5 "drinks mixed with water"
for a total of 13 - however, the drinks mixed with water already
contain one of the flavors, so you have to compute them separately.
Hope that makes sense - for example, a Lemonade couldn't be mixed
with lemon, so for those other 5 drinks, you drop the number of add-ins
to 15.

(Then once you're all done, you have to subtract one for "base
slush" which no one would order by itself, without an add-in and also
add 6 drinks that we sell outright with no addin options such as Apple
Juice, Orange Juice, Iced Tea, etc.)
-------------
Well, I guess that solves it. They had a method after all.

-Taylor Smith

[Jens Kruse Andersen]

Sonic wrote to jtsmith2:

> First, we have 8 "base drinks" and 5 "drinks mixed with water"
> for a total of 13 - however, the drinks mixed with water already
> contain one of the flavors, so you have to compute them separately.
> Hope that makes sense - for example, a Lemonade couldn't be mixed
> with lemon, so for those other 5 drinks, you drop the number of add-ins
> to 15.
>
> (Then once you're all done, you have to subtract one for "base
> slush" which no one would order by itself, without an add-in and also
> add 6 drinks that we sell outright with no addin options such as Apple
> Juice, Orange Juice, Iced Tea, etc.)

That works when the already flavored drinks can also get up to 6 add-ins:
gp > 8*sum(i=0,6,binomial(16,i)) + 5*sum(i=0,6,binomial(15,i)) - 1 + 6
%96 = 168894

Inadequate original info as I suspcted.

--
Jens Kruse Andersen