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Difference between revisions of "2020 AMC 8 Problems/Problem 7"

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First, observe that the second digit of the four digit number cannot be a <math>1</math> or a <math>2</math> because the digits must be distinct and increasing. The second digit also cannot be a <math>4</math> because the number must be less than <math>2400</math>. Thus, the second digit must be <math>3</math>. If we place a <math>4</math> in the third digit then there are 5 ways to select the last digit, namely the last digit could then be <math>5,6,7,8,</math> or <math>9</math>. If we place a <math>5</math> in the third digit, there are 4 ways to select the last digit, namely the last digit could then be <math>6,7,8,</math> or <math>9</math>. Similarly, if the third digit is <math>6</math>, there are 3 ways to select the last digit, etc. Thus, it follows that the total number of valid numbers is <math>5+4+3+2+1=15\implies\boxed{\textbf{(C) }15}</math>.<br>
 
First, observe that the second digit of the four digit number cannot be a <math>1</math> or a <math>2</math> because the digits must be distinct and increasing. The second digit also cannot be a <math>4</math> because the number must be less than <math>2400</math>. Thus, the second digit must be <math>3</math>. If we place a <math>4</math> in the third digit then there are 5 ways to select the last digit, namely the last digit could then be <math>5,6,7,8,</math> or <math>9</math>. If we place a <math>5</math> in the third digit, there are 4 ways to select the last digit, namely the last digit could then be <math>6,7,8,</math> or <math>9</math>. Similarly, if the third digit is <math>6</math>, there are 3 ways to select the last digit, etc. Thus, it follows that the total number of valid numbers is <math>5+4+3+2+1=15\implies\boxed{\textbf{(C) }15}</math>.<br>
 
~ junaidmansuri
 
~ junaidmansuri
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 +
==Solution 2=
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 +
Notice that the number is of the form <math>23AB</math> were <math>A>B>3</math>. We have <math>A=4,B\in [5,9];A=5,B\in [6,9];A=6,B\in [7,9];A=7,B\in [8,9];A=8,B\in [9]</math>. Counting the numbers in the brackets, the answer is <math>5+4+3+2+1=\textbf{(C) }15</math>.
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-franzliszt
  
 
==See also==  
 
==See also==  

Revision as of 11:02, 18 November 2020

Problem 7

How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2347$ is one integer.)

$\textbf{(A) }\text{9} \qquad \textbf{(B) }\text{10} \qquad \textbf{(C) }\text{15} \qquad \textbf{(D) }\text{21}\qquad \textbf{(E) }\text{28}$

Solution 1

First, observe that the second digit of the four digit number cannot be a $1$ or a $2$ because the digits must be distinct and increasing. The second digit also cannot be a $4$ because the number must be less than $2400$. Thus, the second digit must be $3$. If we place a $4$ in the third digit then there are 5 ways to select the last digit, namely the last digit could then be $5,6,7,8,$ or $9$. If we place a $5$ in the third digit, there are 4 ways to select the last digit, namely the last digit could then be $6,7,8,$ or $9$. Similarly, if the third digit is $6$, there are 3 ways to select the last digit, etc. Thus, it follows that the total number of valid numbers is $5+4+3+2+1=15\implies\boxed{\textbf{(C) }15}$.
~ junaidmansuri

=Solution 2

Notice that the number is of the form $23AB$ were $A>B>3$. We have $A=4,B\in [5,9];A=5,B\in [6,9];A=6,B\in [7,9];A=7,B\in [8,9];A=8,B\in [9]$. Counting the numbers in the brackets, the answer is $5+4+3+2+1=\textbf{(C) }15$.

-franzliszt

See also

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png


The thousands place (first digit) has to be a 2 (2020-2400). Since the thousands digit is 2, the next digit must be a 3 (not 4 or onwards because that will go over the range given).

The next digit has to be from 4, 5, 6, 7, or 8. For each of the cases, you get a total of 15 possibilities, which gives you the answer C.


~itsmemasterS

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