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Let S={a,b,c,d,e}. Number of strings of length 5 possible with the letters of S , so that atleast two A's are consecutive is?

Answer given is : 421

1 Answer

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total strings with 5 letters= $5^{5}=3125$

strings with no a's= $4^{5}=1024$

strings with one a's= $5*4^{4}=1280$

strings with two a's which are not consecutive= $6*4^{3}=384$

strings with three a's which are not consecutive = $1*4^{2}=16$

after that with 4 and 5 a's you will get atleast two consecutive a's. So

total 5 length strings with two consecutive a's = $3125-1024-1280-384-16=421$

421 strings are there.

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