DIRECTOR with vowels together: Arrange the letters of DIRECTOR so that all the vowels occur consecutively. How many ways?

Introduction / Context:We keep the vowels together as a block. DIRECTOR has vowels I, E, O (3) and consonants D, R, C, T, R (5, with R repeated twice). Count arrangements of the block with consonants, then the internal permutations of the vowels.

Given Data / Assumptions:Word: D I R E C T O R; vowels = {I, E, O}; consonants = {D, R, C, T, R} with R repeated twice.

Concept / Approach:Treat the vowel block as one entity. Then there are 6 entities (vowel block + 5 consonants), with a duplication among consonants (R×2). Arrange entities and multiply by the permutations within the vowel block.

Step-by-Step Solution:

Verification / Alternative check:Explicit small-case analogs validate the block method with duplicates.

Why Other Options Are Wrong:4320 doubles by ignoring the repeated R; 2720 and 1120 are inconsistent with the duplication and block logic.

Common Pitfalls:Forgetting to divide by 2! for the two R’s, or missing the 3! for the internal vowel permutations.

Final Answer:2160