Difference between revisions of "2020 AMC 8 Problems/Problem 7"
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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>. | 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>. | ||
− | + | -oceanxia | |
-franzliszt | -franzliszt | ||
Revision as of 22:07, 19 November 2020
Problem 7
How many integers between and have four distinct digits arranged in increasing order? (For example, is one integer.)
Solution 1
First, observe that the second digit of the four digit number cannot be a or a because the digits must be distinct and increasing. The second digit also cannot be a because the number must be less than . Thus, the second digit must be . If we place a in the third digit then there are 5 ways to select the last digit, namely the last digit could then be or . If we place a in the third digit, there are 4 ways to select the last digit, namely the last digit could then be or . Similarly, if the third digit is , there are 3 ways to select the last digit, etc. Thus, it follows that the total number of valid numbers is .
~junaidmansuri
Solution 2
Notice that the number is of the form were . We have . Counting the numbers in the brackets, the answer is . -oceanxia -franzliszt
Video Solution
~savannahsolver
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.
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