Question: How many distinct words can be formed using all the letters in the word "BIOINFORMATICS" if the letters must be arranged such that all vowels appear consecutively? - Blask
Title: How Many Distinct Words Can Be Formed from "BIOINFORMATICS" With All Vowels Together?
Title: How Many Distinct Words Can Be Formed from "BIOINFORMATICS" With All Vowels Together?
Introduction
Understanding the Context
The word BIOINFORMATICS is a rich and challenging Latin-derived term packed with consonants and vowels, making it a fascinating puzzle for combinatorial enthusiasts and language lovers alike. A commonly asked question is: How many distinct words can be formed using all the letters in "BIOINFORMATICS" if all vowels must appear together?
In this article, we explore this intriguing linguistic problem from both a mathematical and practical perspective, breaking down the step-by-step process to calculate the number of unique arrangements where all vowels appear as a contiguous block.
Understanding the Word Structure
Image Gallery
Key Insights
The word BIOINFORMATICS contains 16 letters, including:
- Vowels: I, O, I, I, A (5 vowels: I ×3, O ×1, A ×1)
- Consonants: B, N, F, R, M, T, C, S (11 consonants)
Our goal is to arrange all 16 letters such that the five vowels occur together as a single group, preserving their internal order possibilities (since the vowels can repeat), while treating this block as a single “super-letter.”
Step 1: Treat the Vowel Block as a Single Unit
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Because the five vowels must appear consecutively, we treat them as one combined unit or "super-letter." This reduces the problem from arranging 16 letters to arranging:
- 1 vowel block
- 11 consonants
Total objects to arrange = 1 (vowel block) + 11 consonants = 12 items
Step 2: Count Arrangements of the 12 Units
However, these 12 units include both consonants and repeated letters, so we must account for duplicates:
-
Among the 11 consonants: B, N, F, R, M, T, C, S — all distinct
So the consonant collection has no internal repetition. -
The vowel block contains repeated letters: I appears 3 times, O once, A once
Thus, we are arranging 12 positions in which:
- 11 objects (consonants) are distinct
- The vowel block is treated as one unit
- Within the vowel block, the 5 vowels include repetitions
