Difference between revisions of "2019 AMC 10B Problems/Problem 25"

Latest revision as of 19:00, 14 November 2024

The following problem is from both the 2019 AMC 10B #25 and 2019 AMC 12B #23, so both problems redirect to this page.

1 Problem
2 Solution 1 (Recursion)
3 Solution 2 (casework)
4 Solution 3 (casework and blocks)
5 Solution 4 (similar to #3)
6 Solution 5 (Constructive Counting)
7 Solution 6 (Recursion)
8 Solution 7 (Quick Solution by Estimating)
9 Solution 8 (Sequence Dividing and Path Counting)
10 Video Solution by OmegaLearn
11 Video Solution
12 See Also

Problem

How many sequences of $0$ s and $1$ s of length $19$ are there that begin with a $0$ , end with a $0$ , contain no two consecutive $0$ s, and contain no three consecutive $1$ s?

$\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75$

Solution 1 (Recursion)

Let $f(n)$ be the number of valid sequences of length $n$ (satisfying the conditions given in the problem).

We know our valid sequence must end in a $0$ . Then, since we cannot have two consecutive $0$ s, it must end in a $10$ . Now, we only have two cases: it ends with $010$ , or it ends with $110$ which is equivalent to $0110$ . Thus, our sequence must be of the forms $0\ldots010$ or $0\ldots0110$ . In the first case, the first $n-2$ digits are equivalent to a valid sequence of length $n-2$ . In the second, the first $n-3$ digits are equivalent to a valid sequence of length $n-3$ . Therefore, it must be the case that $f(n) = f(n-3) + f(n-2)$ , with $n \ge 3$ (because otherwise, the sequence would contain only 0s and this is not allowed due to the given conditions).

It is easy to find $f(3) = 1$ since the only possible valid sequence is $010$ . $f(4)=1$ since the only possible valid sequence is $0110$ . $f(5)=1$ since the only possible valid sequence is $01010$ .

The recursive sequence is then as follows:

$f(3)=1$ $f(4)=1$ $f(5) = 1$ $f(6) = 1 + 1 = 2$ $f(7) = 1 + 1 = 2$ $f(8) = 1 + 2 = 3$ $f(9) = 2 + 2 = 4$ $f(10) = 2 + 3 = 5$ $f(11) = 3 + 4 = 7$ $f(12) = 4 + 5 = 9$ $f(13) = 5 + 7 = 12$ $f(14) = 7 + 9 = 16$ $f(15) = 9 + 12 = 21$ $f(16) = 12 + 16 = 28$ $f(17) = 16 + 21 = 37$ $f(18) = 21 + 28 = 49$ $f(19) = 28 + 37 = 65$

So, our answer is $\boxed{\text{\bf{(C)} } 65}$ .

Contributors:

~Original Author

~solasky

~BakedPotato66

~CrazyVideoGamez

Solution 2 (casework)

After any particular $0$ , the next $0$ in the sequence must appear exactly $2$ or $3$ positions down the line. In this case, we start at position $1$ and end at position $19$ , i.e. we move a total of $18$ positions down the line. Therefore, we must add a series of $2$ s and $3$ s to get $18$ . There are a number of ways to do this:

Case 1: nine $2$ s - there is only $1$ way to arrange them.

Case 2: two $3$ s and six $2$ s - there are ${8\choose2} = 28$ ways to arrange them.

Case 3: four $3$ s and three $2$ s - there are ${7\choose4} = 35$ ways to arrange them.

Case 4: six $3$ s - there is only $1$ way to arrange them.

Summing the four cases gives $1+28+35+1=\boxed{\textbf{(C) }65}$ .

Solution 3 (casework and blocks)

We can simplify the original problem into a problem where there are $2^{17}$ 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 $0$ s, $1$ s, and $11$ s. Now, we use casework:

Case 1: Alternating 1s and 0s. There is simply 1 way to do this: $0101010101010101010$ . Now, we note that there cannot be only one block of $11$ in the entire sequence, as there must be zeroes at both ends and if we only include 1 block, of $11$ s this cannot be satisfied. This is true for all odd numbers of $11$ blocks.

Case 2: There are 2 $11$ blocks. Using the zeroes in the sequence as dividers, we have a sample as $0110110101010101010$ . We know there are 8 places for $11$ s, which will be filled by $1$ s if the $11$ s don't fill them. This is ${8\choose2} = 28$ ways.

Case 3: Four $11$ blocks arranged. Using the same logic as Case 2, we have ${7\choose4} = 35$ ways to arrange four $11$ blocks.

Case 4: No single $1$ blocks, only $11$ blocks. There is simply one case for this, which is $0110110110110110110$ .

Adding these four cases, we have $1+28+35+1=\boxed{\textbf{(C) }65}$ as our final answer.

~Equinox8

Solution 4 (similar to #3)

Any valid sequence must start with a $0$ . We can then think of constructing a sequence as adding groups of terms to this $0$ , each ending in $0$ . (This is always possible because every valid string ends in $0$ .) For example, we can represent the string $01011010110110$ as: $0-10-110-10-110-110$ . To not have any consecutive 0s, we must have at least one $1$ before the next $0$ . However, we cannot have three or more $1$ s before the next $0$ because we cannot have three consecutive $1$ s. Consequently, we can only have one or two $1$ s.

So we can have the groups: $10$ and $110$ .

After the initial $0$ , we have $18$ digits left to fill in the string. Let the number of $10$ blocks be $x$ , and $110$ be $y$ . Then $x$ and $y$ must satisfy $2x+3y=18$ . We recognize this as a Diophantine equation. Taking $\pmod{2}$ yields $y=0 \pmod{2}$ . Since $x$ and $y$ must both be nonnegative, we get the solutions $(9, 0)$ , $(6, 2)$ , $(3, 4)$ , and $(0, 6)$ . We now handle each of these cases separately.

$(9, 0)$ : Only one arrangement, namely all $10$ s.

$(6, 2)$ : We have 6 groups of $11$ , and $2$ groups of $110$ . This has $\binom{6+2}{2}=28$ cases.

$(3, 4)$ : This means we have 3 groups of $10$ , and 4 groups of $110$ . This has $\binom{3+4}{3}=35$ cases.

$(0, 6)$ : Only one arrangement, namely all $110$ .

Adding these, we have $1+28+35+1=65 \longrightarrow \boxed{(C) 65}$ . ~Math4Life2020

~edited by alpha_2 for spelling and and typos

Solution 5 (Constructive Counting)

Suppose the number of $0$ s is $n$ . We can construct the sequence in two steps:

Step 1: put $n-1$ of $1$ s between the $0$ s;

Step 2: put the rest $19-n-(n-1)=20-2n$ of $1$ s in the $n-1$ spots where there is a $1$ . There are $\binom{n-1}{20-2n}$ ways of doing this.

Now we find the possible values of $n$ :

First of all $n+(n-1) \leq 19 \Rightarrow n\leq 10$ (otherwise there will be two consecutive $0$ s);

And secondly $20-2n \leq n-1\Rightarrow n\geq 7$ (otherwise there will be three consecutive $1$ s).

Therefore the answer is $\sum_{n=7}^{10} \binom{n-1}{20-2n} = \binom{6}{6} + \binom{7}{4} + \binom{8}{2} + \binom{9}{0} = \boxed{\textbf{(C) }65}.$

~ aops

Solution 6 (Recursion)

For a valid sequence of length $n$ , the sequence must be in the form of $01xx...xx10$ . By removing the $01$ at the start of the sequence and the $10$ at the end of the sequence, there are $n-4$ bits left. The $n-4$ 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

So, $f(n) = f(n-4) + 2f(n-5) + f(n-6)$

We will calculate $f(19)$ by Dynamic Programming.

$f(3) = 1$

$f(4) = 1$

$f(5) = 1$

$f(6) = 2$

$f(7) = 2$

$f(8) = 3$

$f(9) = f(5) + 2 \cdot f(4) + f(3) = 1 + 2 \cdot 1 + 1 = 4$

$f(10) = f(6) + 2 \cdot f(5) + f(4) = 2 + 2 \cdot 1 + 1 = 5$

$f(11) = f(7) + 2 \cdot f(6) + f(5) = 2 + 2 \cdot 1 + 1 = 7$

$f(12) = f(8) + 2 \cdot f(7) + f(6) = 3 + 2 \cdot 2 + 2 = 9$

$f(13) = f(9) + 2 \cdot f(8) + f(7) = 4 + 2 \cdot 3 + 2 = 12$

$f(14) = f(10) + 2 \cdot f(9) + f(8) = 5 + 2 \cdot 4 + 3 = 16$

$f(15) = f(11) + 2 \cdot f(10) + f(9) = 7 + 2 \cdot 5 + 4 = 21$

$f(16) = f(12) + 2 \cdot f(11) + f(10) = 9 + 2 \cdot 7 + 5 = 28$

$f(17) = f(13) + 2 \cdot f(12) + f(11) = 12 + 2 \cdot 9 + 7 = 37$

$f(18) = f(14) + 2 \cdot f(13) + f(12) = 16 + 2 \cdot 12 + 9 = 49$

$f(19) = f(15) + 2 \cdot f(14) + f(13) = 21 + 2 \cdot 16 + 12 = \boxed{\text{\bf{(C)} } 65}$

We can further prove $f(n) = f(n-4) + 2f(n-5) + f(n-6)$ is equivalent to $f(n) = f(n-2) + f(n-3)$

Let $k(n) = f(n-2) + f(n-3)$

$k(n-2) = f(n-4) + f(n-5)$

$k(n-3) = f(n-5) + f(n-6)$

$k(n-2)+ k(n-3) = f(n-4) + 2f(n-5) + f(n-6) = f(n)$

$f(n) = k(n-2)+ k(n-3)$

$f(n-2) = k(n-4)+ k(n-5)$

$f(n-3) = k(n-5)+ k(n-6)$

$k(n) = f(n-2) + f(n-3) = k(n-4)+ k(n-5) + k(n-5)+ k(n-6) = k(n-4) + 2k(n-5) + k(n-6)$

So $k(n)$ is the same as $f(n)$ .

~isabelchen

Solution 7 (Quick Solution by Estimating)

Using the tree diagram, you quickly notice that your answer must be very close to a power of $2$ due to the splitting of the tree branches in a tree diagram. $2^6$ is $64$ , which is very close to $65$ , thus giving our answer of $\boxed{\textbf{(C) }65}$ .

~MichaeLLin16 ~Minor Edit and Moving by HappySharks

Solution 8 (Sequence Dividing and Path Counting)

You can split the sequence into $0$ and $6$ strings of three numbers, which can only be $010$ , $011$ , $110$ , $101$ , labelled as A,B,C,D respectively. For example, $0101....010$ is labelled $0DADADA$ . This can also be stated as 0, $D_{1}, A_{2}, D_{3},$ etc. The problem is equivalent to find the number of such sequences. For each string, A can be followed by C and D, B can be followed by A and B, C can be followed by C and D, D can be followed by A,B, and D. If we define $n(A_{i})$ as number of possible sequences that ends with string A at the $i$ th spot (thus be $3i+1$ numbers long), we may find relationship between these numbers by noting that, for the $i+1$ th string, with $i \geq 1$

Since the first number of the sequence must be zero, we must start with $n(A_1) = n(B_1) = 0, n(C_1) = n(D_1) = 1$ . Using the above steps to evolve the situation from i = 1 to i = 6, we get $n(A_{6}) = 37, n(B_{6}) = 37, n(C_{6}) = 28, n(D_{6}) = 49$ . Since the sequence ends with 0, we only need to sum $n(A_{6})$ and $n(C_{6})$ , which yields $37+28=65$ . $\boxed{\textbf{(C) }65}$

Solution by ~Cc2010cc2015

minor edit by ~CLA

Video Solution by OmegaLearn

https://youtu.be/WpSpnx8PPnc?t=94

~pi_is_3.14

Video Solution

https://youtu.be/VamT49PjmdI

@@ Line 1: / Line 1: @@
-{{duplicate|[[2019 AMC 10B Problems|2019 AMC 10B #25]] and [[2019 AMC 12B Problems|2019 AMC 12B #23]]}}
+{{duplicate|[[2019 AMC 10B Problems#Problem 25|2019 AMC 10B #25]] and [[2019 AMC 12B Problems#Problem 23|2019 AMC 12B #23]]}}
 ==Problem==
@@ Line 8: / Line 8: @@
 ==Solution 1 (Recursion)==
-We can deduce, from the given restrictions, that any valid sequence of length <math>n</math> will start with a <math>0</math> followed by either <math>10</math> or <math>110</math>.
+Let <math>f(n)</math> be the number of valid sequences of length <math>n</math> (satisfying the conditions given in the problem).
-Let <math>f(n)</math> 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 <math>n</math>.
-Then we can define a recursive function <math>f(n) = f(n-3) + f(n-2)</math>, with <math>n \ge 3</math> (because otherwise, the sequence would contain only 0s and this is not allowed due to the give conditions).
+We know our valid sequence must end in a <math>0</math>. Then, since we cannot have two consecutive <math>0</math>s, it must end in a <math>10</math>. Now, we only have two cases: it ends with <math>010</math>, or it ends with <math>110</math> which is equivalent to <math>0110</math>. Thus, our sequence must be of the forms <math>0\ldots010</math> or <math>0\ldots0110</math>. In the first case, the first <math>n-2</math> digits are equivalent to a valid sequence of length <math>n-2</math>. In the second, the first <math>n-3</math> digits are equivalent to a valid sequence of length <math>n-3</math>. Therefore, it must be the case that <math>f(n) = f(n-3) + f(n-2)</math>, with <math>n \ge 3</math> (because otherwise, the sequence would contain only 0s and this is not allowed due to the given conditions).
-We derived the recursive function, since for any valid sequence of length <math>n</math>, you can append either <math>10</math> or <math>110</math> to return to the starting position, 0, and the resulting sequence will still satisfy the given conditions.
 It is easy to find <math>f(3) = 1</math> since the only possible valid sequence is <math>010</math>. <math>f(4)=1</math> since the only possible valid sequence is <math>0110</math>. <math>f(5)=1</math> since the only possible valid sequence is <math>01010</math>.
@@ Line 47: / Line 44: @@
 ~BakedPotato66
+~[https://artofproblemsolving.com/wiki/index.php/User:Crazyvideogamez CrazyVideoGamez]
 ==Solution 2 (casework)==
@@ Line 93: / Line 92: @@
 <math>(0, 6)</math>: Only one arrangement, namely all <math>110</math>.
-Adding these, we have <math>1+28+35+1=65 \longrightarrow \boxed{(C)}</math>.
+Adding these, we have <math>1+28+35+1=65 \longrightarrow \boxed{(C) 65}</math>.
 ~Math4Life2020
-~edited by alpha_2 for spelling and and typos (<math>001</math> instead of <math>110</math>)
+~edited by alpha_2 for spelling and and typos
 ==Solution 5 (Constructive Counting)==
@@ Line 116: / Line 115: @@
 </cmath>
-~ asops
+~ aops
+== Solution 6 (Recursion)==
+For a valid sequence of length <math>n</math>, the sequence must be in the form of <math>01xx...xx10</math>. By removing the <math>01</math> at the start of the sequence and the <math>10</math> at the end of the sequence, there are <math>n-4</math> bits left. The <math>n-4</math> bits left can be in the form of:
+ <math>0yy...yy0</math>, the whole <math>(n-4)</math> bits are valid sequence, which is <math>f(n-4)</math>
+ <math>0yy...y01</math>, the <math>(n-5)</math> bits before the last <math>1</math> are valid sequence, which is <math>f(n-5)</math>
+ <math>10y...yy0</math>, the <math>(n-5)</math> bits after the first <math>1</math> are valid sequence, which is <math>f(n-5)</math>
+ <math>10y...y01</math>, the <math>(n-6)</math> bits between the first and last <math>1</math> are valid sequence, which is <math>f(n-6)</math>
+So, <math>f(n) = f(n-4) + 2f(n-5) + f(n-6)</math>
+We will calculate <math>f(19)</math> by [https://en.wikipedia.org/wiki/Dynamic_programming Dynamic Programming].
+<math>f(3) = 1</math>
+<math>f(4) = 1</math>
+<math>f(5) = 1</math>
+<math>f(6) = 2</math>
+<math>f(7) = 2</math>
+<math>f(8) = 3</math>
+<math>f(9) = f(5) + 2 \cdot f(4) + f(3) = 1 + 2 \cdot 1 + 1 = 4</math>
+<math>f(10) = f(6) + 2 \cdot f(5) + f(4) = 2 + 2 \cdot 1 + 1 = 5</math>
+<math>f(11) = f(7) + 2 \cdot f(6) + f(5) = 2 + 2 \cdot 1 + 1 = 7</math>
+<math>f(12) = f(8) + 2 \cdot f(7) + f(6) = 3 + 2 \cdot 2 + 2 = 9</math>
+<math>f(13) = f(9) + 2 \cdot f(8) + f(7) = 4 + 2 \cdot 3 + 2 = 12</math>
+<math>f(14) = f(10) + 2 \cdot f(9) + f(8) = 5 + 2 \cdot 4 + 3 = 16</math>
+<math>f(15) = f(11) + 2 \cdot f(10) + f(9) = 7 + 2 \cdot 5 + 4 = 21</math>
+<math>f(16) = f(12) + 2 \cdot f(11) + f(10) = 9 + 2 \cdot 7 + 5 = 28</math>
+<math>f(17) = f(13) + 2 \cdot f(12) + f(11) = 12 + 2 \cdot 9 + 7 = 37</math>
+<math>f(18) = f(14) + 2 \cdot f(13) + f(12) = 16 + 2 \cdot 12 + 9 = 49</math>
+<math>f(19) = f(15) + 2 \cdot f(14) + f(13) = 21 + 2 \cdot 16 + 12 = \boxed{\text{\bf{(C)}  } 65}</math>
+We can further prove <math>f(n) = f(n-4) + 2f(n-5) + f(n-6)</math> is equivalent to <math>f(n) = f(n-2) + f(n-3)</math>
+Let <math>k(n) = f(n-2) + f(n-3)</math>
+<math>k(n-2) = f(n-4) + f(n-5)</math>
+<math>k(n-3) = f(n-5) + f(n-6)</math>
+<math>k(n-2)+ k(n-3) = f(n-4) + 2f(n-5) + f(n-6) = f(n)</math>
+<math>f(n) = k(n-2)+ k(n-3)</math>
+<math>f(n-2) = k(n-4)+ k(n-5)</math>
+<math>f(n-3) = k(n-5)+ k(n-6)</math>
+<math>k(n) = f(n-2) + f(n-3) = k(n-4)+ k(n-5) + k(n-5)+ k(n-6) = k(n-4) + 2k(n-5) + k(n-6)</math>
+So <math>k(n)</math> is the same as <math>f(n)</math>.
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+== Solution 7 (Quick Solution by Estimating) ==
+Using the tree diagram, you quickly notice that your answer must be very close to a power of <math>2</math> due to the splitting of the tree branches in a tree diagram. <math>2^6</math> is <math>64</math>, which is very close to <math>65</math>, thus giving our answer of <math>\boxed{\textbf{(C) }65}</math>.
+~MichaeLLin16 ~Minor Edit and Moving by [https://artofproblemsolving.com/wiki/index.php/User:HappySharks HappySharks]
+== Solution 8 (Sequence Dividing and Path Counting) ==
+You can split the sequence into <math>0</math> and <math>6</math> strings of three numbers, which can only be <math>010</math>,<math>011</math>,<math>110</math>,<math>101</math>, labelled as A,B,C,D respectively. For example, <math>0101....010</math> is labelled <math>0DADADA</math>. This can also be stated as 0, <math>D_{1}, A_{2}, D_{3},</math> etc. The problem is equivalent to find the number of such sequences. For each string, A can be followed by C and D, B can be followed by A and B, C can be followed by C and D, D can be followed by A,B, and D.  If we define <math>n(A_{i})</math> as number of possible sequences that ends with string A at the <math>i</math>th spot (thus be <math>3i+1</math> numbers long), we may find relationship between these numbers by noting that, for the <math>i+1</math> th string, with <math>i \geq 1</math>
+ <math>n(A_{i+1}) = n(B_{i+1}) = n(B_{i}) + n(D_{i}), n(C_{i+1}) = n(A_{i}) + n(C_{i})</math>
+ <math>n(D_{i+1}) =  n(A_{i}) + n(C_{i}) + n(D_{i})</math>.
+Since the first number of the sequence must be zero, we must start with <math>n(A_1) = n(B_1) = 0, n(C_1) = n(D_1) = 1</math>. Using the above steps to evolve the situation from i = 1 to i = 6, we get <math>n(A_{6}) = 37, n(B_{6}) = 37, n(C_{6}) = 28, n(D_{6}) = 49</math>.
+Since the sequence ends with 0, we only need to sum <math>n(A_{6})</math> and <math>n(C_{6})</math>, which yields <math>37+28=65</math>.  <math>\boxed{\textbf{(C) }65}</math>
+Solution by ~Cc2010cc2015
+minor edit by ~CLA
+== Video Solution by OmegaLearn ==
+https://youtu.be/WpSpnx8PPnc?t=94
+~pi_is_3.14
 ==Video Solution==
-For those who want a video solution: https://youtu.be/VamT49PjmdI
+https://youtu.be/VamT49PjmdI
 ==See Also==
@@ Line 125: / Line 216: @@
 {{AMC12 box|year=2019|ab=B|num-b=22|num-a=24}}
 {{MAA Notice}}
+[[Category: Intermediate Combinatorics Problems]]

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