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

The word INSURANCE contains 9 letters, including the 4 vowels I, U, A, E and the 5 consonants N, S, R, N, C. Treating the vowels as a single block to keep them together, we now have 6 units to arrange, which can be done in 6! / 2! = 360 ways due to the repeated N. The 4 vowels within their block can be arranged in 4! = 24 ways. Multiplying these arrangements gives 360 * 24 = 8640 total ways.

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