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Prove the pattern


In drawing lowercase English letters a through z uniformly at random, the expected time till you see willywilly is 26^10 + 26^5.
In drawing numerals over 0-9 uniformly at random, the expected time till you see 1231231 is 10^7 + 10^4 + 10.
In flipping a fair coin, the expected time till you see HTHTTHT is 2^7 + 2^2.
See a pattern? Find it ... then prove it.
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