Discussion

In how many ways can we arrange the word ‘FUZZTONE’ so that all the vowels come together?

Answer: C

FUZZTONE has eight letters with Z appearing twice: vowels U, O, E and consonants F, Z, Z, T, N.

Treat the three vowels as one block, giving six units among which Z repeats:

\[ \frac{6!}{2!} = 360 \]

Within the block the three distinct vowels arrange in

\[ 3! = 6 \text{ ways} \]

\[ 360 \times 6 = 2160 \]

The repeated Z is divided out among the units, but the vowels are all different so no division is needed inside the block.

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