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In how many ways the letters of the word "INSURANCE" be arranged so that the vowels never occur together?

This is how I did the problem,

Since number of permutations of n different things taken all at a time, when m specified things come together is \((n!-m!)\times (n-m+1)!\)

Number of ways the vowels never occur together = \((9!-4!)\times (9-4+1)!\)

which gives a value greater than the total number of permutations with that word!!!

How is it possible?Is the formula wrong?

Another way is to find answer is,

Number of ways vowels never occur together=Total permutations - No.of permutations in which vowels occur together =\(\frac { 9! }{ 2! } -\frac { 6!\times 4! }{ 2! } =172800\)

Is my second way correct?

Note by Anandhu Raj
1 year, 11 months ago

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Also it is not stated that n and m are all distinct or not if you use the formula you can't take n=9 Saumya Acharya · 1 year, 11 months ago

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@Saumya Acharya May be!!:) Anandhu Raj · 1 year, 11 months ago

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The Second way is unquestionably correct. Satyen Nabar · 1 year, 11 months ago

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@Satyen Nabar What about the first one? Whether such a formula exist(seen it in a book)? Anandhu Raj · 1 year, 11 months ago

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@Anandhu Raj No but if u have its a misprint. Satyen Nabar · 1 year, 11 months ago

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@Satyen Nabar Thanks :) Anandhu Raj · 1 year, 11 months ago

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