In how many other ways can the letters of the word MULTIPLE be arranged,
(i) without changing the order of the vowels
(ii) keeping the position of each vowel fixed &
(iii) without changing the relative order/position of vowels & consonants.

Answer

To solve the problem of arranging the letters of the word "MULTIPLE" under the specified conditions, we will break it down into three parts as per the question.

### Step 1: Understanding the Composition of the Word
The word "MULTIPLE" consists of 8 letters: M, U, L, T, I, P, L, E. Among these, the vowels are U, I, E, and the consonants are M, L, T, P, L.

### Step 2: Total Arrangements of the Word
First, we calculate the total arrangements of the letters in "MULTIPLE". Since the letter L appears twice, the formula for arrangements is given by:

\[ \text{Total arrangements} = \frac{n!}{p!} \]

where \( n \) is the total number of letters, and \( p \) is the factorial of the number of times any letter repeats.

Thus, we have:

\[ \text{Total arrangements} = \frac{8!}{2!} = \frac{40320}{2} = 20160 \]

### (i) Arrangements without Changing the Order of the Vowels
To find the arrangements without changing the order of the vowels (U, I, E), we treat the vowels as fixed in their positions. The vowels must always appear in the order U, I, E.

1. Identify the positions of the vowels: U, I, E can only occupy their respective positions.
2. The remaining letters (M, L, T, P, L) can be arranged in the remaining positions.

The arrangement of the consonants is:

\[ \text{Arrangements of consonants} = \frac{5!}{2!} = \frac{120}{2} = 60 \]

Thus, the total arrangements without changing the order of the vowels is:

\[ \text{Total} = 60 \]

### (ii) Keeping the Position of Each Vowel Fixed
In this case, we keep the vowels fixed in their original positions (U, I, E) and only rearrange the consonants (M, L, T, P, L) in the remaining positions.

The number of arrangements of the consonants remains the same as calculated above:

\[ \text{Arrangements} = \frac{5!}{2!} = 60 \]

### (iii) Without Changing the Relative Order of Vowels and Consonants
Here, we need to maintain the relative order of vowels and consonants. The vowels (U, I, E) must maintain their order, but we can rearrange the consonants (M, L, T, P, L) in such a way that they still appear in the same relative positions.

1. The total arrangements of the consonants is still:

\[ \text{Arrangements of consonants} = \frac{5!}{2!} = 60 \]

2. Since the vowels can be arranged in 3! ways, we multiply this by the arrangements of consonants:

\[ \text{Total arrangements} = 60 \times 3! = 60 \times 6 = 360 \]

### Final Answers
- (i) Arrangements without changing the order of the vowels: **60**
- (ii) Keeping the position of each vowel fixed: **60**
- (iii) Without changing the relative order of vowels and consonants: **360**