An ice cream parlor sell ice creams with three different flavors: blue- berry, chocolates, and vanilla. How many ways are there to choose a dozen ice creams with at most six vanilla ice creams? (Note: one dozen consists of twelve ice creams).

- anonymous

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- anonymous

I would suggest this method:
How many ways are there with exactly 0 vanilla
How many ways are there with exactly 1 vanilla
How many ways are there with exactly 2 vanilla
How many ways are there with exactly 3 vanilla
...
How many ways are there with exactly 6 vanilla
Then I would add them up.

- anonymous

To find how many ways with n vanilla, we ask:
How many ways to distribute 12-n ice creams amongst the 2 other flavors

- anonymous

Perhaps something like this?\[ \Large
\sum_{n=0}^{6} 2^{12-n}
\]

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## More answers

- anonymous

Hmmm. See any flaws?

- anonymous

so 1) if a dozen consist the same flavor? chocolate and blueberry = 2 ways
2) if vanilla = 6, so x+ y = 12 , x for chocolates and y for bluberry?
until if vanilla = 0 , so x+y = 12 ?

- anonymous

2) if vanilla = 6, so x+ y = 6 , x for chocolates and y for bluberry? until if vanilla = 0 , so x+y = 12 ?

- anonymous

You don't wanna think in terms of x+y=6. We aren't solving for the number of chocolates and blueberry. We are finding the total number of combinations.

- anonymous

Can I ask you some questions? You can answer me the best you can.

- anonymous

Suppose I gave you a choice between chocolate ice cream and blueberry ice cream...
How many choices do you have?

- anonymous

I mean options, how many options do you have?

- anonymous

yeah, i think if x+y = 6 , with times and divide theory, (6+1)! / 6!1! ?

- anonymous

Nope, just answer my questions ok? It'll help you get to the solution.

- anonymous

If I ask you if you want chocolate or blueberry, how many option do you have?

- anonymous

It's not a trick question or anything, you have only 2 options in this case.

- anonymous

13 ?

- anonymous

if must a dozen, right?

- anonymous

If you are going to get 2 ice creams, and you get to chose a flavor for each, how many options do you have?

- anonymous

(We are starting out small and working our way up)

- anonymous

oooh , if how many flavors?

- anonymous

There are 2 flavors, and 2 cones. You get to chose a flavor for each cone. How many possibilities are there?

- anonymous

3 ?

- anonymous

yeah, cause order doesn't matter right?

- anonymous

ohyeaaah, i got that

- anonymous

What about if there are n cones.... how many options?

- anonymous

if there are 3 cones... how many options?

- anonymous

n+1 ?

- anonymous

but i think, in my case, the flavor has given,

- anonymous

Sure.

- anonymous

so, we must think about the order

- anonymous

If you have 6 cones, how many possibilities with 2 flavors?

- anonymous

But having a blueberry and a chocolate isn't any different than having a chocolate and a blueberry... this is what is meant by order doesn't matter.

- anonymous

i thinks its different, if 2 blueberry and 1 chocolates, with 2 chocolates and 1 blueberry

- anonymous

oke, can give me a time, some bussiness

- anonymous

7+8+9+10+11+12+13=70

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