MHB Find a recursive formula for the number of combinations

AI Thread Summary
The discussion focuses on finding a recursive formula for the number of valid combinations of words of length 'n' using the letters A, B, and C, while avoiding specific combinations. The recursive formula established is a_n = 2 + a_{n-1}, where a_1 = 3 represents the initial condition with three valid single-letter words. The reasoning behind a_{n-1} is clarified by noting that if the first letter is C, there are no restrictions on the subsequent letters, effectively reducing the problem to finding combinations for n-1 letters. This understanding helps bridge the gap in grasping recursion in combinatorial contexts. The discussion emphasizes the importance of recognizing how initial conditions and recursive relationships define the overall count of valid combinations.
CStudent
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I have some question integrating between combinatorics and recursive formulas.Generally, I have some difficulty with the concept of recursion, as well as with the recursion in programming unfortunately.I have some question to solve, and maybe you can guide me:Find a recursive formula and a terminal condition for the number of words with the length 'n' that can be written by $A, B, C$, such that these combinations won't be shown: $AB, AC, BA, BC$.Now, I know that if we start from $A$ or $B$ we of course have one option, 'n' $A's$, 'n' $B's$ - which are two combinations.Now, if we start from $C$, I understand we have more 'n-1' letters to write such that we don't write the illegal ones.I know that means we have $a_{n-1}$ combinations after $C$.
But that is what I don't understand, what is that $a_{n-1}$, how do we know it really contains only the legal combinations?I wold love to get some sense of this concept.Thanks.
 
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CStudent said:
I have some question integrating between combinatorics and recursive formulas.Generally, I have some difficulty with the concept of recursion, as well as with the recursion in programming unfortunately.I have some question to solve, and maybe you can guide me:Find a recursive formula and a terminal condition for the number of words with the length 'n' that can be written by $A, B, C$, such that these combinations won't be shown: $AB, AC, BA, BC$.Now, I know that if we start from $A$ or $B$ we of course have one option, 'n' $A's$, 'n' $B's$ - which are two combinations.Now, if we start from $C$, I understand we have more 'n-1' letters to write such that we don't write the illegal ones.I know that means we have $a_{n-1}$ combinations after $C$.
But that is what I don't understand, what is that $a_{n-1}$, how do we know it really contains only the legal combinations?I wold love to get some sense of this concept.Thanks.
If $a_n$ is the number of permissible words of length $n$, then you have shown that $a_n = 2 + a_{n-1}$. (In fact, there is only one word starting with A, one word starting with B, and $a_{n-1}$ words starting with C.) That is the recursive formula. To complete the definition, you need to add that $a_1 = 3$. I'm guessing that this is what the question means by a terminal condition, though I would call it the initial condition. It simply says that there are three words of length 1, namely A, B and C.
 
Opalg said:
If $a_n$ is the number of permissible words of length $n$, then you have shown that $a_n = 2 + a_{n-1}$. (In fact, there is only one word starting with A, one word starting with B, and $a_{n-1}$ words starting with C.) That is the recursive formula. To complete the definition, you need to add that $a_1 = 3$. I'm guessing that this is what the question means by a terminal condition, though I would call it the initial condition. It simply says that there are three words of length 1, namely A, B and C.

Yes, but I have an obstacle in understanding the concept of the recursion...

Again, I don't understand how we can leap to $a_{n−1}$. Okay, after the first C, we have n-1 places, how does it tells us that has $a_{n−1}$? How do we know what is this $a_{n−1}$ at all?

Thank you.
 
CStudent said:
Yes, but I have an obstacle in understanding the concept of the recursion...

Again, I don't understand how we can leap to $a_{n−1}$. Okay, after the first C, we have n-1 places, how does it tells us that has $a_{n−1}$? How do we know what is this $a_{n−1}$ at all?

Thank you.
If the first letter is a C then there is no restriction on what the next letter can be. In other words, it's exactly equivalent to starting again, but with only $n-1$ places instead of $n$.
 
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