2012 AMC 12B Problems/Problem 12
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
There are selections; however, we count these twice, therefore
. The wording of the question implies D, not E.
However, MAA decided to accept both D and E.
Consider the 20 term sequence of 's and 's. Keeping all other terms 1, a sequence of consecutive 0's can be placed in locations. That is, there are 20 strings with 1 zero, 19 strings with 2 consecutive zeros, 18 strings with 3 consecutive zeros, ..., 1 string with 20 consecutive zeros. Hence there are strings with consecutive zeros. The same argument shows there are strings with consecutive 1's. This yields strings in all. However, we have counted twice those strings in which all the 1's and all the 0's are consecutive. These are the cases , , , ..., (of which there are 19) as well as the cases , , , ..., (of which there are 19 as well). This yields
First, we think of ways to make all the 's consecutive. If there are no consecutive 's, there are ways to order them. If there is one consecutive , there are ways to order them. If there are two consecutive 's, then there are ways to order them (We treat the two 's like a block, and then order that block with 18 other 's). Continuing in this fashion, there are ways to order consecutive 's. From symmetry, there are also ways to order the 's. Now, from PIE, we subtract out the cases where both the 's and the 's are consecutive. We do this because when counting the ways to order the 's, we counted all of these cases once. Then, we did so again when ordering the 's. So, to only have all of these cases once, we must subtract them. If is the leftmost digit, then there are cases where all the 's and 's are consecutive (we basically are choosing how many 's are consecutive, and there are possibilities. All other digits become , which are automatically consecutive since the 's are consecutive. There are also cases when is the left-most digit. Thus, there are a total of . But, from the way the problem is worded, it somewhat implies that the orderings must include both 's and 's, so the answer would then be after we subtract out the cases where the orderings are either all 's or all 's. But, since this is unclear, MAA accepted both and as acceptable answers.
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