To find the number of ways to arrange all the letters of the word 'COMBINATIONS' around a circle such that no two vowels are together, let's break it down step-by-step.
Identify Vowels and Consonants:
Vowels: 0, O, I, I, A (total of 5 vowels)
Consonants: C, M, B, N, T, N, S (total of 7 consonants)
Arrangement of Consonants:
Since the arrangement is in a circle, we arrange the consonants in
(7−1)!=6 ! ways because in circular permutations, we fix one position to break the circle and linearly order the rest.
Interleaving Vowels:
Arranging the consonants creates 7 gaps (before the first consonant, between each pair, and after the last consonant) to place the vowels. We need to select 5 gaps for the vowels, ensuring no two vowels are adjacent, which is crucial since we want them not together.
Arrangement of Vowels:
Arrange the vowels
0,0,I,I,A in these gaps.
The arrangement of the vowels is affected by repetition. The formula for arranging ' n ' items where there are repetitions is given by
‌, where
ki represents the number of times each repeated letter occurs.
The necessary calculation for the vowels:
‌ because 0 and I each repeat twice.
Total Arrangements:
Combine the arrangements of consonants and vowels as follows:
‌ Total ways ‌=6!×‌By further expanding this calculation based on the repetitions:
‌This gives the total number of arrangements of the letters in 'COMBINATIONS' around a circle, ensuring that no two vowels are adjacent.