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Difference between revisions of "2012 AMC 12B Problems/Problem 12"

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==Solution 3==
 
==Solution 3==
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First, we think of ways to make all the <math>1</math>'s consecutive. If there are no consecutive <math>1</math>'s, there are <math>\binom{20}{0}</math> ways to order them. If there is one consecutive <math>1</math>, there are <math>\binom{20}{1}</math> ways to order them. If there are two consecutive <math>1</math>'s, then there are <math>\binom{19}{1}</math> ways to order them (We treat the two <math>1</math>'s like a block, and then order that block with 18 other <math>0</math>'s). Continuing in this fashion, there are <math>\binom{20}{0} + \binom{20}{1} + \binom{19}{1} + \cdots + \binom{1}{1} = 1 + 20 + 19 + \cdots + 2 + 1 = 210 + 1 = 211</math> ways to order consecutive <math>1</math>'s. From symmetry, there are also <math>211</math> ways to order the <math>0</math>'s. Now, from PIE, we subtract out the cases where both the <math>1</math>'s and the <math>0</math>'s are consecutive. We do this because when counting the ways to order the <math>1</math>'s, we counted all of these cases once. Then, we did so again when ordering the <math>0</math>'s. So, to only have all of these cases once, we must subtract them. If <math>1</math> is the leftmost digit, then there are <math>20</math> cases where all the <math>1</math>'s and <math>0</math>'s are consecutive (we basically are choosing how many <math>1</math>'s are consecutive, and there are <math>20</math> possibilities. All other digits become <math>0</math>, which are automatically consecutive since the <math>1</math>'s are consecutive. There are also <math>20</math> cases when <math>0</math> is the left-most digit. Thus, there are a total of <math>211 + 211 - 20 - 20 = \boxed{\textbf{(E)}\ 382}</math>. But, from the way the problem is worded, it somewhat implies that the orderings must include both <math>1</math>'s and <math>0</math>'s, so the answer would then be <math>\boxed{\textbf{(D)}\ 380</math> after we subtract out the cases where the orderings are either all <math>1</math>'s or all <math>0</math>'s. But, since this is unclear, MAA accepted both <math>\boxed{\textbf{(D}\ 380}</math> and <math>\boxed{\textbf{(E)}\ 382}</math> as acceptable answers.
+
First, we think of ways to make all the <math>1</math>'s consecutive. If there are no consecutive <math>1</math>'s, there are <math>\binom{20}{0}</math> ways to order them. If there is one consecutive <math>1</math>, there are <math>\binom{20}{1}</math> ways to order them. If there are two consecutive <math>1</math>'s, then there are <math>\binom{19}{1}</math> ways to order them (We treat the two <math>1</math>'s like a block, and then order that block with 18 other <math>0</math>'s). Continuing in this fashion, there are <math>\binom{20}{0} + \binom{20}{1} + \binom{19}{1} + \cdots + \binom{1}{1} = 1 + 20 + 19 + \cdots + 2 + 1 = 210 + 1 = 211</math> ways to order consecutive <math>1</math>'s. From symmetry, there are also <math>211</math> ways to order the <math>0</math>'s. Now, from PIE, we subtract out the cases where both the <math>1</math>'s and the <math>0</math>'s are consecutive. We do this because when counting the ways to order the <math>1</math>'s, we counted all of these cases once. Then, we did so again when ordering the <math>0</math>'s. So, to only have all of these cases once, we must subtract them. If <math>1</math> is the leftmost digit, then there are <math>20</math> cases where all the <math>1</math>'s and <math>0</math>'s are consecutive (we basically are choosing how many <math>1</math>'s are consecutive, and there are <math>20</math> possibilities. All other digits become <math>0</math>, which are automatically consecutive since the <math>1</math>'s are consecutive. There are also <math>20</math> cases when <math>0</math> is the left-most digit. Thus, there are a total of <math>211 + 211 - 20 - 20 = \boxed{\textbf{(E)}\ 382}</math>. But, from the way the problem is worded, it somewhat implies that the orderings must include both <math>1</math>'s and <math>0</math>'s, so the answer would then be <math>\boxed{\textbf{(D)}\ 380}</math> after we subtract out the cases where the orderings are either all <math>1</math>'s or all <math>0</math>'s. But, since this is unclear, MAA accepted both <math>\boxed{\textbf{(D}\ 380}</math> and <math>\boxed{\textbf{(E)}\ 382}</math> as acceptable answers.
  
 
== See Also ==
 
== See Also ==

Revision as of 07:46, 29 June 2023

Problem

How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?

$\textbf{(A)}\ 190\qquad\textbf{(B)}\ 192\qquad\textbf{(C)}\ 211\qquad\textbf{(D)}\ 380\qquad\textbf{(E)}\ 382$

Solutions

Solution 1

There are $\binom{20}{2}$ selections; however, we count these twice, therefore

$2\cdot\binom{20}{2} = \boxed{\textbf{(D)}\ 380}$. The wording of the question implies D, not E.

However, MAA decided to accept both D and E.

Solution 2

Consider the 20 term sequence of $0$'s and $1$'s. Keeping all other terms 1, a sequence of $k>0$ consecutive 0's can be placed in $21-k$ locations. That is, there are 20 strings with 1 zero, 19 strings with 2 consecutive zeros, 18 strings with 3 consecutive zeros, ..., 1 string with 20 consecutive zeros. Hence there are $20+19+\cdots+1=\binom{21}{2}$ strings with consecutive zeros. The same argument shows there are $\binom{21}{2}$ strings with consecutive 1's. This yields $2\binom{21}{2}$ strings in all. However, we have counted twice those strings in which all the 1's and all the 0's are consecutive. These are the cases $01111...$, $00111...$, $000111...$, ..., $000...0001$ (of which there are 19) as well as the cases $10000...$, $11000...$, $111000...$, ..., $111...110$ (of which there are 19 as well). This yields $2\binom{21}{2}-2\cdot19=\boxed{\textbf{(E)}\ 382}$

Solution 3

First, we think of ways to make all the $1$'s consecutive. If there are no consecutive $1$'s, there are $\binom{20}{0}$ ways to order them. If there is one consecutive $1$, there are $\binom{20}{1}$ ways to order them. If there are two consecutive $1$'s, then there are $\binom{19}{1}$ ways to order them (We treat the two $1$'s like a block, and then order that block with 18 other $0$'s). Continuing in this fashion, there are $\binom{20}{0} + \binom{20}{1} + \binom{19}{1} + \cdots + \binom{1}{1} = 1 + 20 + 19 + \cdots + 2 + 1 = 210 + 1 = 211$ ways to order consecutive $1$'s. From symmetry, there are also $211$ ways to order the $0$'s. Now, from PIE, we subtract out the cases where both the $1$'s and the $0$'s are consecutive. We do this because when counting the ways to order the $1$'s, we counted all of these cases once. Then, we did so again when ordering the $0$'s. So, to only have all of these cases once, we must subtract them. If $1$ is the leftmost digit, then there are $20$ cases where all the $1$'s and $0$'s are consecutive (we basically are choosing how many $1$'s are consecutive, and there are $20$ possibilities. All other digits become $0$, which are automatically consecutive since the $1$'s are consecutive. There are also $20$ cases when $0$ is the left-most digit. Thus, there are a total of $211 + 211 - 20 - 20 = \boxed{\textbf{(E)}\ 382}$. But, from the way the problem is worded, it somewhat implies that the orderings must include both $1$'s and $0$'s, so the answer would then be $\boxed{\textbf{(D)}\ 380}$ after we subtract out the cases where the orderings are either all $1$'s or all $0$'s. But, since this is unclear, MAA accepted both $\boxed{\textbf{(D}\ 380}$ and $\boxed{\textbf{(E)}\ 382}$ as acceptable answers.

See Also

2012 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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All AMC 12 Problems and Solutions

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