Question

In how many ways can the letters of the word $INSURANCE$ be arranged, so that the vowels are never separate?

• A. $4320$
• B. $5570$
• C. $7020$
• D. $8640$

Answer 1

The correct option is D $8640$

Note that there are $4$ vowels $(I,U,A,E)$ in the word and $5$ consonants $(N,S,R,N,C)$ of which ${}^{\prime }{N}^{\prime }$ is repeated twice.
Now, we want all the vowels together, hence group them as a one unit.to form the permutations.
Hence we have to arrange total of $6$ units ($1$ group of vowel and $5$ consonants - one of them is repeated)
This can be done in $\frac{6!}{2!}$.
But the $4$ vowels in the group can be arrange among themselves in $4!$ ways for each of the above arrangement,
Hence the total number of possible arrangement are $\frac{6!}{2!}\times 4!=\frac{720\times 24}{2}=8640$