2012 AMC 10B Problems/Problem 22
Let (, , ... ) be a list of the first 10 positive integers such that for each either or or both appear somewhere before in the list. How many such lists are there?
If we have 1 as the first number, then the only possible list is .
If we have 2 as the first number, then we have 9 ways to choose where the goes, and the numbers ascend from the first number, , with the exception of the . For example, , or . There are ways to do so.
If we use 3 as the first number, we need to choose 2 spaces to be 2 and 1, respectively. There are ways to do this.
In the same way, the total number of lists is:
By the binomial theorem, this is = , or
Arrange the spaces and put arrows pointing either up or down between them. Then for each arrangement of arrows there is one and only one list that corresponds to up. For example, all arrows pointing up is . There are 9 arrows, so the answer is =
NOTE: Solution cited from: http://www.artofproblemsolving.com/Videos/external.php?video_id=269.
Notice that the answer to the problem is solely based on the length of the lists, i.e. 10. We can replace 10 with smaller values, such as 2 and 3, and try to find a pattern. If we replace it with 2, we can easily see that there are two possible lists, and . If we replace it with 3, there are four lists, and . Since 2 and 4 are both powers of 2, it is likely that the number of lists is , where is the length of the lists.
Solution 4 (Recursion)
If , the sequence must be . If , then , and the sequence is . If , then the possible sequences are In general, for an -length sequence, if , then through can be filled in ways with through , and through must be sorted in decreasing order with the remaining numbers ( through ), in one way. Thus , where .
We can see (or prove by induction) that . Hence, .
Assume the same conditions to be held and let's look at several smaller cases to find a pattern. If we are only arranging there are trivially only ways. Now let us look at arranging . You can arrange this in ways. Looking at you can arrange this in ways. The pattern becomes evident now. If there are numbers there are ways. Hence our answer would be ways which is .
Solution 3 and 5 states that without formal proof. Solution 4 gives a formal proof. Here is another formal proof:
. When the list goes from numbers to numbers, there are ways to make the new lists:
Case 1: append to the end of lists with numbers to make a new list, the number of the new lists is ;
Case 2: put number at the end of the new lists, the way to arrange as the first items is the same as to arrange , by subtracting 1 from each of the elements, so the number of the new lists is also .
Video Solution by Richard Rusczyk
Video Solution by TheBeautyofMath
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