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Number of permutations of the word PANCH...

Number of permutations of the word PANCHKULA where A and U are separated. The word PANCHKULA must be separated.

Text Solution

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Case I : All A, U, A are separated
First arrange P, N, C, H, K, L in 6!.
Seven gaps are formed .
`xx P xx Nxx Cxx Hxx KxxLxx`
A,U, A are arranged in these gaps.
Three gaps can be selected in `.^(7)C_(3)` ways.
In these three gaps A, U, A can be arranged in `(3!)/(2!)=3` ways.
So, number of words in this case are `=6!xx .^(7)C_(3)xx3`
Case II : A A are together but separated from U.
Here we need two gaps for A A and U.
So, number of words in this case are `6!xx .^(7)C_(2)xx2!`
From above cases total number of words
`=6!xx .^(7)C_(3)xx3+6!xx .^(7)C_(2)xx2!`
`=21xx7!`
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