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

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==Solution 1==
 
==Solution 1==
 
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>
~jmansuri
+
~ junaidmansuri
  
 
==See also==  
 
==See also==  

Revision as of 06:25, 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

See also

2020 AMC 8 (ProblemsAnswer KeyResources)
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

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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