The number of ways in which the letters of the word MULTIPLE be arranged without changing the order of the vowels is
There are 3 vowels, so the 5 consonants can be divided into groups of
0,0,0,5
0,0,1,4
0,0,2,3
...
5,0,0,0
For each of those groups i,j,k,l the consonants can be permuted in
i! * j! * k! * l! ways
Then divide by 2, since the 2 L's are indistinguishable.
Maybe someone else can think of a simpler way.
To find the number of ways in which the letters of the word "MULTIPLE" can be arranged without changing the order of the vowels, we can follow these steps:
Step 1: Identify the vowels
The vowels in the word "MULTIPLE" are U, I, and E.
Step 2: Fix the position of vowels
Since we want to keep the vowels in their original order, we first need to fix their positions. In this case, U, I, and E already have fixed positions.
Step 3: Arrange the consonants
Now, we can arrange the consonants (M, L, T, P) among themselves without changing the order of the vowels.
There are 4 consonants, so we have 4 available positions. We can arrange them in 4! (4 factorial) ways, which is equal to 4 x 3 x 2 x 1 = 24.
Step 4: Calculate the total number of arrangements
Since the vowels are already fixed, and the consonants can be arranged in 24 ways, we multiply the fixed vowel arrangement (1) by the consonant arrangements (24) to get the total number of arrangements:
Total number of arrangements = 1 x 24 = 24
Therefore, the number of ways to arrange the letters of the word "MULTIPLE" without changing the order of the vowels is 24.