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?

## Comments

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TopNewestAlso 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 · 2 years, 1 month ago

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– Anandhu Raj · 2 years, 1 month ago

May be!!:)Log in to reply

The Second way is unquestionably correct. – Satyen Nabar · 2 years, 1 month ago

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– Anandhu Raj · 2 years, 1 month ago

What about the first one? Whether such a formula exist(seen it in a book)?Log in to reply

– Satyen Nabar · 2 years, 1 month ago

No but if u have its a misprint.Log in to reply

– Anandhu Raj · 2 years, 1 month ago

Thanks :)Log in to reply