Difference between revisions of "2017 AMC 10B Problems/Problem 17"

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Thus our final answer is <math>511+1022-9 = \boxed{\textbf{(B) } 1524}</math>.
 
Thus our final answer is <math>511+1022-9 = \boxed{\textbf{(B) } 1524}</math>.
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==Solution 3==
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Unlike the first two solutions, we can do our casework based off of whether or not the number contains a <math>0</math>.
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If it does, then we know the <math>0</math> must be the last digit in the number (and hence, the number has to be decreasing). Because <math>0</math> is not positive, <math>0</math> is not monotonous. So, we need to pick at least <math>1</math> number in the set <math>[1, 9].</math> After choosing our numbers, there will be just <math>1</math> way to arrange them so that the overall number is monotonous.
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In total, each of the <math>9</math> digits can either be in the monotonous number or not, yielding <math>2^9 = 512</math> total solutions. However, we said earlier that <math>0</math> cannot be by itself so we have to subtract out the case in which we picked none of the numbers <math>1-9</math>. So, this case gives us <math>511</math>.
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Onto the second case, if there are no <math>0</math>s, then the number can either be arranged in ascending order or in descending order. So, for each selection of the digits <math>1- 9</math>, there are <math>2</math> solutions. This gives <cmath>2 \cdot (2^9 - 1) = 2 \cdot 511 = 1022</cmath> possibilities. Note that we subtracted out the <math>1</math> because we cannot choose none of the numbers.
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However, realize that if we pick just <math>1</math> digit, then there is only <math>1</math> arrangement. We cannot put a single digit in both ascending and descending order. So, we must subtract out <math>9</math> from there (because there are <math>9</math> possible ways to select one number and for each case, we overcounted by <math>1</math>).
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All in all, that gives <math>511 + 1022 - 9 = \boxed{(B) 1524}</math> monotonous numbers.
  
 
==See Also==
 
==See Also==

Revision as of 13:26, 29 December 2020

The following problem is from both the 2017 AMC 12B #11 and 2017 AMC 10B #17, so both problems redirect to this page.

Problem

Call a positive integer $monotonous$ if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there?

$\textbf{(A)}\ 1024\qquad\textbf{(B)}\ 1524\qquad\textbf{(C)}\ 1533\qquad\textbf{(D)}\ 1536\qquad\textbf{(E)}\ 2048$

Solution 1

Case 1: monotonous numbers with digits in ascending order

There are $\Sigma_{n=1}^{9} \binom{9}{n}$ ways to choose n digits from the digits 1 to 9. For each of these ways, we can generate exactly one monotonous number by ordering the chosen digits in ascending order. Note that 0 is not included since it will always be a leading digit and that is not allowed. Also, $\emptyset$ (the empty set) isn't included because it doesn't generate a number. The sum is equivalent to $\Sigma_{n=0}^{9} \binom{9}{n} -\binom{9}{0} = 2^9 - 1 = 511.$

Case 2: monotonous numbers with digits in descending order

There are $\Sigma_{n=1}^{10} \binom{10}{n}$ ways to choose n digits from the digits 0 to 9. For each of these ways, we can generate exactly one monotonous number by ordering the chosen digits in descending order. Note that 0 is included since we are allowed to end numbers with zeros. However, $\emptyset$ (the empty set) still isn't included because it doesn't generate a number. The sum is equivalent to $\Sigma_{n=0}^{10} \binom{10}{n} -\binom{10}{0} = 2^{10} - 1 = 1023.$ We discard the number 0 since it is not positive. Thus there are $1022$ here.

Since the 1-digit numbers 1 to 9 satisfy both case 1 and case 2, we have overcounted by 9. Thus there are $511+1022-9=\boxed{\textbf{(B)}  1524}$ monotonous numbers.

Solution 2

Like Solution 1, divide the problem into an increasing and decreasing case:

$\bullet$ Case 1: Monotonous numbers with digits in ascending order.

Arrange the digits 1 through 9 in increasing order, and exclude 0 because a positive integer cannot begin with 0.

To get a monotonous number, we can either include or exclude each of the remaining 9 digits, and there are $2^9 = 512$ ways to do this. However, we cannot exclude every digit at once, so we subtract 1 to get $512-1=511$ monotonous numbers for this case.

$\bullet$ Case 2: Monotonous numbers with digits in descending order.

This time, we arrange all 10 digits in decreasing order and repeat the process to find $2^{10} = 1024$ ways to include or exclude each digit. We cannot exclude every digit at once, and we cannot include only 0, so we subtract 2 to get $1024-2=1022$ monotonous numbers for this case.

At this point, we have counted all of the single-digit monotonous numbers twice, so we must subtract 9 from our total.

Thus our final answer is $511+1022-9 = \boxed{\textbf{(B) } 1524}$.

Solution 3

Unlike the first two solutions, we can do our casework based off of whether or not the number contains a $0$.

If it does, then we know the $0$ must be the last digit in the number (and hence, the number has to be decreasing). Because $0$ is not positive, $0$ is not monotonous. So, we need to pick at least $1$ number in the set $[1, 9].$ After choosing our numbers, there will be just $1$ way to arrange them so that the overall number is monotonous.

In total, each of the $9$ digits can either be in the monotonous number or not, yielding $2^9 = 512$ total solutions. However, we said earlier that $0$ cannot be by itself so we have to subtract out the case in which we picked none of the numbers $1-9$. So, this case gives us $511$.


Onto the second case, if there are no $0$s, then the number can either be arranged in ascending order or in descending order. So, for each selection of the digits $1- 9$, there are $2$ solutions. This gives \[2 \cdot (2^9 - 1) = 2 \cdot 511 = 1022\] possibilities. Note that we subtracted out the $1$ because we cannot choose none of the numbers.

However, realize that if we pick just $1$ digit, then there is only $1$ arrangement. We cannot put a single digit in both ascending and descending order. So, we must subtract out $9$ from there (because there are $9$ possible ways to select one number and for each case, we overcounted by $1$).

All in all, that gives $511 + 1022 - 9 = \boxed{(B) 1524}$ monotonous numbers.

See Also

2017 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2017 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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