The number of ways is 720.

Step 1: Understand the word “LANGUAGE”

Write the letters and count each:

  • L – 1
  • A – 2
  • N – 1
  • G – 1
  • U – 1
  • E – 1

So:

  • Total letters = 8
  • Vowels = A, A, U, E → 4 vowels (with A repeated twice)
  • Consonants = L, N, G → 3 consonants

We need arrangements where all vowels stay together.

Step 2: Treat all vowels as one block

Imagine the 4 vowels “A A U E” as one super-letter (a block). Now the objects to arrange are:

  • [VOWEL-BLOCK]
  • L
  • N
  • G

So we have 4 objects in total. Number of ways to arrange these 4 distinct objects:

4!=244!=244!=24

Step 3: Arrange the vowels inside their block

Inside the vowel-block we have: A, A, U, E If all four were different, there would be 4!=244!=244!=24 arrangements,
but there are two identical A’s , so we divide by 2!2!2!:

Arrangements of vowels=4!2!=242=12\text{Arrangements of vowels}=\frac{4!}{2!}=\frac{24}{2}=12Arrangements of vowels=2!4!​=224​=12

Step 4: Combine both parts

Total valid arrangements =
(arrangements of the 4 objects) × (arrangements of vowels within block)

=4!×4!2!=24×12=288=4!\times \frac{4!}{2!}=24\times 12=288=4!×2!4!​=24×12=288

Wait – that suggests 288, but we must double-check the consonant count carefully. Recheck letters: L, A, N, G, U, A, G, E Notice: G appears twice , not once. Correct counts:

  • L – 1
  • A – 2
  • N – 1
  • G – 2
  • U – 1
  • E – 1

So:

  • Vowels: A, A, U, E → 4 letters (A repeated twice)
  • Consonants: L, N, G, G → 4 letters (G repeated twice)
  • Total letters = 8 (still fine)

Now, treat all vowels as one block: Objects to arrange:

  • [VOWEL-BLOCK]
  • L
  • N
  • G
  • G

That is 5 objects , with G repeated twice. Number of ways to arrange these 5 objects:

5!2!=1202=60\frac{5!}{2!}=\frac{120}{2}=602!5!​=2120​=60

Arrangements of vowels inside the block are still:

4!2!=12\frac{4!}{2!}=122!4!​=12

So the correct total :

Total ways=60×12=720\text{Total ways}=60\times 12=720Total ways=60×12=720

Final Answer

In how many ways can the letters of the word “LANGUAGE” be arranged so that the vowels always come together? Answer: 720 ways. Bottom note: Information gathered from public forums or data available on the internet and portrayed here.