AMC 8 2012 Problem 16
AMC8 2012 年第 16 题
基本计数原理★★★☆☆
Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
用数字 0,1,2,3,4,5,6,7,8,9 组成两个 5 位数（每个数字只使用一次），使这两个五位数的和最大。下面哪个可能是这两个数之一？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
In order to maximize the sum of the numbers, the numbers must have their digits ordered in decreasing value. There are only two numbers from the answer choices with this property: 76531 and 87431. To determine the answer we will have to use estimation and the first two digits of the numbers. For 76531 the number that would maximize the sum would start with 98. The first two digits of 76531 (when rounded) are 77. Adding 98 and 77, we find that the first three digits of the sum of the two numbers would be 175. For 87431 the number that would maximize the sum would start with 96. The first two digits of 87431 (when rounded) are 87. Adding 96 and 87, we find that the first three digits of the sum of the two numbers would be 183. From the estimations, we can say that the answer to this problem is 87431. p.s. USE INTUITION, see answer choices before solving any question -litttle_master
In order to determine the largest number possible, we have to evenly distribute the digits when adding. The two numbers that show an example of this are 97531 and 86420. The digits can be interchangeable between numbers because we only care about the actual digits. The first digit must be either 9 or 8. This immediately knocks out . The second digit must be either 7 or 6. This doesn't cancel any choices. The third digit must be either 5 or 4. This knocks out and . The fourth digit must be 3 or 2. This cancels out . This leaves us with 87431.
If we use intuition, we know that the digits will be IN ORDER, to maximze the number. This eliminates , ,and . Additionally, the 2 two numbers must have 9 or 8 for the first digit to maximize the sum, eliminating . This leaves 87431. -written by litttle_master purely, not copied from anywhere
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