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Question: Bob makes a list of numbers, where each number is 1, 2, or 3, and the number 2 appears an even number of times. How many different lists could Bob have made?

 

I tried looking for patterns, but so far I haven't had much luck. Help would be appreciated!

 May 20, 2021
 #1
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The wording is kind of confusing but shouldn't he be able to make infinite number of combinations if he can just list them over and over?

 May 20, 2021
 #2
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That is correct. However, the question is asking in terms of n.

 

So, for example, if n = 1, then there are 2 options: 1 or 3

If n = 2, then there are 5 options: (11), (22), (33), (13), or (31)

 

We need to find some sort of recursion that works for any number n I think but I'm not sure how.

 May 20, 2021
 #3
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After some messing around, I got this:

\(\displaystyle \sum_{k=0}^{\operatorname{floor}\left(\frac{n}{2}\right)+1}\operatorname{nCr}\left(n,\ 2k\right)\cdot2^{\left(n-2k\right)}\)

Basically, the floor(n/2)+1 calculates the number of possible twos given n. For example, if n is 3, the number of possible twos is floor(3/2)+1=2 (which is correct, since there could be either 0 or 2 twos)

The nCr(n, 2k) basically chooses the spots for the twos to be in.

Lastly, the 2^(n-2k) fills the rest of the remaining spots with either 3s or 1s. 

It does not look very neat, but I don't really have the effort and determination to reduce it though lol

textot  May 21, 2021

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