Difference between revisions of "2020 AMC 8 Problems/Problem 7"
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==Solution 3== | ==Solution 3== | ||
Distinct Digits means the numbers are in increasing order, but it cannot repeat the same note. For example, 1234 and 1235 can be a distinct digit, while 1223 cannot. So, the answers are 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, and 2389. This means 15 distinct digits from 2020 to 2400. Therefore, the answer is <math>\boxed{\textbf{(C) }15}</math>. | Distinct Digits means the numbers are in increasing order, but it cannot repeat the same note. For example, 1234 and 1235 can be a distinct digit, while 1223 cannot. So, the answers are 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, and 2389. This means 15 distinct digits from 2020 to 2400. Therefore, the answer is <math>\boxed{\textbf{(C) }15}</math>. | ||
+ | ~HappyGenius1124😉 | ||
==Video Solution by WhyMath== | ==Video Solution by WhyMath== |
Revision as of 18:13, 15 January 2022
Contents
Problem
How many integers between and
have four distinct digits arranged in increasing order? (For example,
is one integer.)
Solution 1
Firstly, observe that the second digit of such a number cannot be or
, because the digits must be distinct and increasing. The second digit also cannot be
as the number must be less than
, so it must be
. It remains to choose the latter two digits, which must be
distinct digits from
. That can be done in
ways; there is then only
way to order the digits, namely in increasing order. This means the answer is
.
Solution 2 (without using the "choose" function)
As in Solution 1, we find that the first two digits must be , and the third digit must be at least
. If it is
, then there are
choices for the last digit, namely
,
,
,
, or
. Similarly, if the third digit is
, there are
choices for the last digit, namely
,
,
, and
; if
, there are
choices; if
, there are
choices; and if
, there is
choice. It follows that the total number of such integers is
.
Solution 3
Distinct Digits means the numbers are in increasing order, but it cannot repeat the same note. For example, 1234 and 1235 can be a distinct digit, while 1223 cannot. So, the answers are 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, and 2389. This means 15 distinct digits from 2020 to 2400. Therefore, the answer is .
~HappyGenius1124😉
Video Solution by WhyMath
~savannahsolver
Video Solution
https://youtu.be/61c1MR9tne8 ~ The Learning Royal
Video Solution by Interstigation
https://youtu.be/YnwkBZTv5Fw?t=251
~Interstigation
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.