Question

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).

Answer 1

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.

Answer 2

To find how many ways with n vanilla, we ask:

How many ways to distribute 12-n ice creams amongst the 2 other flavors

Answer 3

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

Answer 4

Hmmm. See any flaws?

Answer 5

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 ?

Answer 6

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

Answer 7

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.

Answer 8

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

Answer 9

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

Answer 10

I mean options, how many options do you have?

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

13 ?

Answer 16

if must a dozen, right?

Answer 17

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

Answer 18

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

Answer 19

oooh , if how many flavors?

Answer 20

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

Answer 21

3 ?

Answer 22

yeah, cause order doesn't matter right?

Answer 23

ohyeaaah, i got that

Answer 24

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

Answer 25

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

Answer 26

n+1 ?

Answer 27

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

Answer 28

Sure.

Answer 29

so, we must think about the order

Answer 30

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

Answer 31

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.

Answer 32

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

Answer 33

oke, can give me a time, some bussiness

Answer 34

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