alleryIn how many ways can the letters of the word 'TRIANGLE' be arranged such that all the VOWELS are TOGETHER?
A$8!$
B$3! \cdot 6!$
C$3! \cdot 5!$
D$\dfrac{8!}{3!}$
Answer & Solution
Correct answer: B. $3! \cdot 6!$
1. TRIANGLE has 8 letters: T, R, I, A, N, G, L, E. Vowels: I, A, E (3 vowels). Consonants: T, R, N, G, L (5 consonants).
2. STEP 1: Treat all 3 vowels as a SINGLE BLOCK. Now we have 6 objects: 5 consonants + 1 vowel-block.
3. Arrange these 6 objects: $6!$ ways.
4. STEP 2: The 3 vowels INSIDE the block can be arranged among themselves in $3!$ ways.
5. By the multiplication principle: total = $3! \cdot 6! = 6 \cdot 720 = 4320$ arrangements.
6. Option A counts ALL arrangements without the constraint. Option C uses 5! incorrectly. Option D doesn't model the problem.
_Source: syllabus Mathematics, Ch 6, §6.3 + Examples with grouping constraints, p. 7–8._
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