2017 AMC 10 B
Complete problem set with solutions and individual problem pages
Call a positive integer monotonous if it is a onedigit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, ,, and are monotonous, but ,, and are not. How many monotonous positive integers are there? (2017 AMC 10B Problem, Question#17)
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Case : monotonous numbers with digits in ascending order.
There are ways to choose n digits from the digits to . For each of these ways, we can generate exactly one monotonous number by ordering the chosen digits in ascending order. Note that 0 is not included since it will always be a leading digit and that is not allowed. Also, (the empty set) isn't included because it doesn't generate a number. The sum is equivalent to .
Case : monotonous numbers with digits in descending order.
There are ways to choose n digits from the digits to . For each of these ways, we can generate exactly one monotonous number by ordering the chosen digits in descending order. Note that is included since we are allowed to end numbers with zeros. However, (the empty set) still isn't included because it doesn't generate a number. The sum is equivalent to .
We discard the number since it is not positive. Thus there are here.
Since the digit numbers to satisfy both case and case , we have overcounted by . Thus there are monotonous numbers.
Like Solution , divide the problem into an increasing and decreasing case:
Case : Monotonous numbers with digits in ascending order.
Arrange the digits through in increasing order, and exclude because a positive integer cannot begin with .
To get a monotonous number, we can either include or exclude each of the remaining digits, and there are ways to do this. However, we cannot exclude every digit at once, so we subtract to get monotonous numbers for this case.
Case : Monotonous numbers with digits in descending order.
This time, we arrange all digits in decreasing order and repeat the process to find ways to include or exclude each digit. We cannot exclude every digit at once, and we cannot include only , so we subtract to get monotonous numbers for this case. At this point, we have counted all of the single-digit monotonous numbers twice, so we must subtract from our total.
Thus our final answer is .
