It concerns to permutation 'n combination exercises... the question is: If you have 32 different flavours, How many different double scoop ice cream cones can you make with two flavours? a) Must...

It concerns to permutation 'n combination exercises...

the question is: If you have 32 different flavours, How many different double scoop ice cream cones can you make with two flavours?

a) Must be different flavors      the answer is:496

b) Must be the same flavor      the answer is:328

The problem is the procedure... I don't get it...

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tjbrewer | Elementary School Teacher | (Level 2) Associate Educator

Posted on

The procedure is a simple method of combinatorics.

Lets say you only had 3 flavors: Chocolate (Ch), Vanilla (Va), and Pistachio (Ps), how many different double scoop Ice cream cones can you make with two flavors.  This means that each pair of scoops must be different flavors. 

a) the order of flavors does matter

b) the order of flavors doesn't matter

In a) for each flavor we put on the bottom scoop, there are two flavors that we can put on top of it; combinatorics, multiplies the number of choices we have for the first scoop (3) by the number of choices for the second scoop (2) to get 6 different combinations as demonstrated:

Ch-Va, Ch-Ps,
Va-Ch, Va-Ps,
Ps-Va, Ps-Ch,

However, in b) Ch-Va, and Va-Ch are the same combination of flavors.  So are Ch-Ps, and Ps-Ch; and Ps-Va, and Va-Ps.  This eliminates 3 of the combinations you can have.  So you only have 3 different ice-cream combinations. 

So with your assortment of 32 flavors: 32 choices for the first scoop times 31 choices for the second scoop yields should yield 992 different combinations with a different order of scoops (i.e. permutations).

If the order of each pairing doesn't matter then there is a ratio of 2:1 Permutations:Combinations.  Applying that ratio to our found permutations.  `992:x=2:1 x=992-:2=496`

` ` Your answers should be a)992, and b)496. 

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