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I have a question that needs some explaining and it has to do with two sentences that are supposedly asking different things, but to me I can't seem to find out why both sentences are different

Question 1: How many ways are there for 12 ice cream cones to have all same flavors out of 31 flavors? Answer to this is obviously 31

VS.

Question 2: How many ways are there if a flavor may be ordered as many as 12 times (31 total diff flavors to choose from)? Answer is (31+12-1,12)<<<< combinations with repetitions

Can someone explain to me how these two questions are different because in my eyes I can't seem to see what the difference is..My english comprehension in math is way below average >.>..I keep reading these two over and over again and I cna't seem to see why these two are asking different things...

Ray
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2 Answers2

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You are to order 12 cones of ice cream and have 31 flavours to choose from. In the first question, you must order the same flavour for all twelve cones (e.g. 12 times strawberry, or 12 times vanilla, or twelve times banana, or ...). In the second question you are free to order any flavours you want, with or without repetition, but you do not distinguish the cones (i.e. eight cones strawberry, three cones vanilla, one cone banana is the same as three cones vanilla, five cones strawberry, one cone vanilla and three more cones of strawberry)

Hagen von Eitzen
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Q1. How many ways are there for 12 ice cream cones to have all same flavors out of 31 flavors?

This means all cones have same flavour. 
And since we have 31 different flavours to choose from, the answer is obviously 31.


Question 2: How many ways are there if a flavor may be ordered as many as 12 times (31 total diff flavors to choose from)?

Here, each flavour can be ordered 0, 1, 2, .... 12 times to fill each cone.
this can be done in (n+m-1, m) ways.
An example with smaller numbers: flavours: 3 (ABC) and cones:2
combinations:
repeating: AA/BB/CC
non repeating: AB/BC/CA

(3+2-1,2)=(4,2) = 6

You can refer to this link.

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tpb261
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