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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).

Discrete Math
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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.
To find how many ways with n vanilla, we ask: How many ways to distribute 12-n ice creams amongst the 2 other flavors
Perhaps something like this?\[ \Large \sum_{n=0}^{6} 2^{12-n} \]

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Other answers:

Hmmm. See any flaws?
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 ?
2) if vanilla = 6, so x+ y = 6 , x for chocolates and y for bluberry? until if vanilla = 0 , so x+y = 12 ?
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.
Can I ask you some questions? You can answer me the best you can.
Suppose I gave you a choice between chocolate ice cream and blueberry ice cream... How many choices do you have?
I mean options, how many options do you have?
yeah, i think if x+y = 6 , with times and divide theory, (6+1)! / 6!1! ?
Nope, just answer my questions ok? It'll help you get to the solution.
If I ask you if you want chocolate or blueberry, how many option do you have?
It's not a trick question or anything, you have only 2 options in this case.
13 ?
if must a dozen, right?
If you are going to get 2 ice creams, and you get to chose a flavor for each, how many options do you have?
(We are starting out small and working our way up)
oooh , if how many flavors?
There are 2 flavors, and 2 cones. You get to chose a flavor for each cone. How many possibilities are there?
3 ?
yeah, cause order doesn't matter right?
ohyeaaah, i got that
What about if there are n cones.... how many options?
if there are 3 cones... how many options?
n+1 ?
but i think, in my case, the flavor has given,
Sure.
so, we must think about the order
If you have 6 cones, how many possibilities with 2 flavors?
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.
i thinks its different, if 2 blueberry and 1 chocolates, with 2 chocolates and 1 blueberry
oke, can give me a time, some bussiness
7+8+9+10+11+12+13=70

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