Inside Out

Probability Level 3

Within Riley's mind, there are five manifestations of her emotions - Joy, Sadness, Anger, Digust and Fear. Her new memories are stored in colored orbs in her conscious mind. If there are different colored orbs for different combination of 1 or more of her 5 emotions, how many distinct colored orbs are there?

As an explicit example: there can have a bluish yellow orb with emotions of Joy and Sadness; there can also have a reddish purplish greenish blue orb with emotions of Anger, Fear, Disgust and Sadness.

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The answer is 31.

2 solutions

Vishnu Bhagyanath
Aug 20, 2015

Riley would have $\binom{5}{1}$ orbs having a single emotion, $\binom{5}{2}$ of a combination of two emotions and so on. Therefore, in total, Riley would have $\binom{5}{1}+\binom{5}{2}+\binom{5}{3}+\binom{5}{4}+\binom{5}{5} = \boxed{31}$

ALTERNATE SOLUTION :

An easier way is to understand that $\sum_{i=0}^{n} \binom {n}{i} = 2^n$ Therefore, $\sum_{i=1}^{5} \binom {5}{i} = 2^5 -1 =31$

Pi Han Goh - 5 years, 9 months ago

Awesome solution.Upvoted!

Athiyaman Nallathambi - 5 years, 9 months ago

Jan Hrček
Aug 20, 2015

Each orb corresponds to a nonempty subset of the five emotions. In general $n$ element set has $2^n$ subsets, including the empty set, thus there are $2^5-1 = 31$ orbs corresponding to the subsets.

Moderator note:

That's right. Nice use of applying subsets into your solution!