Question

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is________

Answer 1

Correct Answer: 50400

The correct answer is 50400
Vowels : A,A,A,I,I,O
Consonants : S,S,S,S,N,N,T
$\because$ Total number of ways in which vowels come together
$=\frac{\lfloor 8}{\lfloor 4\lfloor 2}\times \frac{\lfloor 6}{\lfloor 3\lfloor 2}=50400$

Answer 2

To solve this problem, we need to find the number of permutations of the word "ASSASSINATION" such that all the vowels (A, A, A, I, I, O) occur together.
Step-by-Step Solution:
1. Identify the Vowels and Consonants:
  - Vowels: A, A, A, I, I, O
  - Consonants: S, S, S, S, N, T, N
2. Treat the Group of Vowels as a Single Entity:
  Since all vowels need to occur together, we can treat the group of vowels as a single entity. Let this group be represented as \( V \).
3. Count the Number of Entities:
  Now we have the following entities: \( V \), S, S, S, S, N, T, N. This gives us a total of 8 entities.
4. Permutations of the Entities:
  The total number of permutations of these 8 entities, considering the repetition of the consonants, is given by:
  \[ \frac{8!}{4! \cdot 2!}\]
  Here, \( 4! \) accounts for the repetition of the 4 S's and \( 2! \) accounts for the repetition of the 2 N's.
5. Calculate the Permutations:
  \[  8! = 40320  \]
  \[ 4! = 24 \]
  \[  2! = 2\]
  \[ \frac{8!}{4! \cdot 2!} = \frac{40320}{24 \cdot 2} = \frac{40320}{48} = 840  \]
6. Permutations of the Vowels within the Group:
  Now we need to consider the permutations of the vowels within the group \( V \). The number of permutations of the vowels A, A, A, I, I, O is:
  \[\frac{6!}{3! \cdot 2!}  \]
  Here, \( 3! \) accounts for the repetition of the 3 A's and \( 2! \) accounts for the repetition of the 2 I's.
7. Calculate the Permutations of Vowels:
  \[  6! = 720\]
  \[ 3! = 6\]
  \[ 2! = 2  \]
  \[ \frac{6!}{3! \cdot 2!} = \frac{720}{6 \cdot 2} = \frac{720}{12} = 60 \]
8. Total Number of Permutations:
  The total number of permutations where the vowels occur together is the product of the permutations of the entities and the permutations of the vowels within the group:
  \[ 840 \times 60 = 50400 \]
Conclusion:
The number of words that can be formed using all the letters of the word "ASSASSINATION" such that the vowels occur together is \( 50400 \).

Answer 3

Grouping all vowels together as a single unit: \[ \text{Vowels: A, A, A, I, I, O} \] \[ \text{Consonants: S, S, S, S, N, N, T} \] Total number of ways vowels occur together: \[ = \frac{8!}{4!2!} \times \frac{6!}{3!2!} = 50400 \]