Arranging Letters of EQUATION: Vowels and Consonants Together
This problem involves finding the number of ways to arrange the letters of the word 'EQUATION' under a specific condition: all the vowels must stay together, and all the consonants must also stay together.
Identifying Vowels and Consonants
First, let's identify the vowels and consonants in the word EQUATION:
- The word EQUATION has 8 distinct letters.
- Vowels: E, U, A, I, O. There are 5 vowels.
- Consonants: Q, T, N. There are 3 consonants.
Grouping Vowels and Consonants Together
The condition requires that all vowels are grouped together and all consonants are grouped together. We can think of these groups as single units:
- Let the block of vowels be represented as [VVVVV].
- Let the block of consonants be represented as [CCC].
Now, we need to arrange these two blocks. The possible arrangements of these blocks are:
- [VVVVV] [CCC]
- [CCC] [VVVVV]
There are 2 possible ways to arrange these two blocks.
Calculating Arrangements Within Each Group
Next, we calculate how many ways the letters can be arranged within each block:
- Arranging the Vowels: The 5 vowels (E, U, A, I, O) can be arranged among themselves in \(5!\) ways.
Calculation: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) ways. - Arranging the Consonants: The 3 consonants (Q, T, N) can be arranged among themselves in \(3!\) ways.
Calculation: \(3! = 3 \times 2 \times 1 = 6\) ways.
Determining the Total Number of Arrangements
To find the total number of arrangements satisfying the condition, we use the multiplication principle. We multiply the number of ways to arrange the blocks by the number of ways to arrange the letters within each block:
Total Arrangements = (Ways to arrange the 2 blocks) \(\times\) (Ways to arrange vowels within their block) \(\times\) (Ways to arrange consonants within their block)
Total Arrangements = \(2! \times 5! \times 3!\)
Total Arrangements = \(2 \times 120 \times 6\)
Total Arrangements = \(1440\)
Therefore, there are 1440 distinct arrangements of the letters of the word EQUATION where all vowels are together and all consonants are together.
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