Let's analyze the problem step-by-step.
Problem:
How many ways can the letters of the word "language" be arranged such that all the vowels always come together?
Step 1: Identify the letters and vowels
The word "language" has 8 letters:
- Letters: L, A, N, G, U, A, G, E
- Vowels: A, A, U, E
- Consonants: L, N, G, G
Step 2: Treat all vowels as a single unit
Since all vowels must come together, consider the group of vowels (A, A, U, E) as one single unit. Now, the units to arrange are:
- The vowel block (V)
- L
- N
- G
- G
So, we have 5 units: V, L, N, G, G
Step 3: Arrange the units
The number of ways to arrange these 5 units, considering the repetition of G twice, is:
5!2!=1202=60\frac{5!}{2!}=\frac{120}{2}=602!5!=2120=60
Step 4: Arrange the vowels inside the vowel block
The vowels inside the block are A, A, U, E. Number of ways to arrange these 4 vowels, considering the repetition of A twice, is:
4!2!=242=12\frac{4!}{2!}=\frac{24}{2}=122!4!=224=12
Step 5: Calculate total arrangements
Multiply the arrangements of the units by the arrangements of vowels inside the block:
60×12=72060\times 12=72060×12=720
Final answer:
There are 720 ways to arrange the letters of the word "language" such that all vowels come together.
