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Suppose Elmo starts with the string "SesameStreet" and then applies one modification to it, where a modification consists of inserting a number at some position. This process can produce a string like "Sesa5meStreet". What if Elmo applies two modifications? This process can produce a string like "9Sesa5meStreet". How many different strings can Elmo create? |
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There are $13$ "interletter gaps" in SesameStreet, of which $11$ are genuinely between letters, and the other $2$ are at the two ends. We can either (i) choose $1$ of these gaps, and put in it any string of two digits or (ii) choose $2$ of these gaps, and insert a digit into the leftmost chosen gap, and then a digit into the other gap. For type (i), there are $\binom{13}{1}$ ways to choose the gap, and for each such way we can enter any of the $(10)(10)$ sequences of two not necessarily distinct digits. For type (ii), there are $\binom{13}{2}$ ways to choose the two gaps, and for each way there are $(10)(10)$ ways to fill the two gaps with digits. So we get a total of $$\binom{13}{1}(100)+\binom{13}{2}(100)$$ ways to do the job. Calculate. We have $\binom{13}{1}=13$ and $\binom{13}{2}=78$. This gives a total of $9100$. |
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There are thirteen places to insert a number and we need two of them (possibly choosing the same place twice). This known as a combination with repetition. In our case, there are $$ \binom{13 + 2 - 1}{2} = \binom{14}{2} = 91 $$ ways to choose the locations for the modifications. For each such choice of positions, there are $10$ choices for the first modification and $10$ choices for the second, giving $10 \cdot 10 = 100$ modifications total. Finally, we have $$ 91 \cdot 100 = 9,100 $$ total possible modifications. |
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