2019 AMC 10B Problems/Problem 25
- The following problem is from both the 2019 AMC 10B #25 and 2019 AMC 12B #23, so both problems redirect to this page.
How many sequences of s and s of length are there that begin with a , end with a , contain no two consecutive s, and contain no three consecutive s?
Solution 1 (Recursion)
We can deduce, from the given restrictions, that any valid sequence of length will start with a followed by either or . Let be the number of valid (meaning: the sequence contains 0s and 1s, starts and ends with both 0, and there are no two consecutive 0s and no three consecutive 1s) sequences of length .
Then we can define a recursive function , with (because otherwise, the sequence would contain only 0s and this is not allowed due to the give conditions).
We derived the recursive function, since for any valid sequence of length , you can append either or to return to the starting position, 0, and the resulting sequence will still satisfy the given conditions.
It is easy to find since the only possible valid sequence is . since the only possible valid sequence is . since the only possible valid sequence is .
The recursive sequence is then as follows:
So, our answer is .
Solution 2 (casework)
After any particular , the next in the sequence must appear exactly or positions down the line. In this case, we start at position and end at position , i.e. we move a total of positions down the line. Therefore, we must add a series of s and s to get . There are a number of ways to do this:
Case 1: nine s - there is only way to arrange them.
Case 2: two s and six s - there are ways to arrange them.
Case 3: four s and three s - there are ways to arrange them.
Case 4: six s - there is only way to arrange them.
Summing the four cases gives .
Solution 3 (casework and blocks)
We can simplify the original problem into a problem where there are binary characters with zeros at the beginning and the end. Then, we know that we cannot have a block of 2 zeroes and a block of 3 ones. Thus, our only options are a block of s, s, and s. Now, we use casework:
Case 1: Alternating 1s and 0s. There is simply 1 way to do this: . Now, we note that there cannot be only one block of in the entire sequence, as there must be zeroes at both ends and if we only include 1 block, of s this cannot be satisfied. This is true for all odd numbers of blocks.
Case 2: There are 2 blocks. Using the zeroes in the sequence as dividers, we have a sample as . We know there are 8 places for s, which will be filled by s if the s don't fill them. This is ways.
Case 3: Four blocks arranged. Using the same logic as Case 2, we have ways to arrange four blocks.
Case 4: No single blocks, only blocks. There is simply one case for this, which is .
Adding these four cases, we have as our final answer.
Solution 4 (similar to #3)
Any valid sequence must start with a . We can then think of constructing a sequence as adding groups of terms to this , each ending in . (This is always possible because every valid string ends in .) For example, we can represent the string as: . To not have any consecutive 0s, we must have at least one before the next . However, we cannot have three or more s before the next because we cannot have three consecutive s. Consequently, we can only have one or two s.
So we can have the groups: and .
After the initial , we have digits left to fill in the string. Let the number of blocks be , and be . Then and must satisfy . We recognize this as a Diophantine equation. Taking yields . Since and must both be nonnegative, we get the solutions , , , and . We now handle each of these cases separately.
: Only one arrangement, namely all s.
: We have 6 groups of , and groups of . This has cases.
: This means we have 3 groups of , and 4 groups of . This has cases.
: Only one arrangement, namely all .
Adding these, we have . ~Math4Life2020
~edited by alpha_2 for spelling and and typos
Solution 5 (Constructive Counting)
Suppose the number of s is . We can construct the sequence in two steps:
Step 1: put of s between the s;
Step 2: put the rest of s in the spots where there is a . There are ways of doing this.
Now we find the possible values of :
First of all (otherwise there will be two consecutive s);
And secondly (otherwise there will be three consecutive s).
Therefore the answer is
Solution 6 (Recursion)
For a valid sequence of length , the sequence must be in the form of . By removing the at the start of the sequence and the at the end of the sequence, there are bits left. The bits left can be in the form of:
, the whole bits are valid sequence, which is , the bits before the last are valid sequence, which is , the bits after the first are valid sequence, which is , the bits between the first and last are valid sequence, which is
We will calculate by Dynamic Programming.
We can further prove is equivalent to
So is the same as .
Video Solution by OmegaLearn
For those who want a video solution: https://youtu.be/VamT49PjmdI
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