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Find the number of permutations of lette...

Find the number of permutations of letters `a ,n ,c ,d ,e ,f,g` taken all together if neither `b egnorc a d` pattern appear.

Text Solution

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Total number of permutations without any restrictions are, n(U)=7!
Number of permutations in which beg (beg) acdf pattern always appear are, n(A)=5!
Number of permutations in which 'cad' (cad) befg pattern always appear are, n(B)=5!.
Number of permutation in which both 'beg' (cad)(beg) f and 'cad' pattern appear are, `n(A cap B)=3!`.
Then number of permutations in which 'beg' and 'cad' pattern do not appear
`=n(A' cap B')`
`=n(U)-n(A cup B)`
`=n(U)-[n(A)+n(B)-n(A cap B)]`
`=7!-[5!+5!-3!]`
=4806
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