consider the word MULTIPLE then in how many ways letters of word MULTIPLE can be arranged
A) without changing the order of vowels equals
B) keeping the position of each vowel fixed equals
C) without changing the relative order / position of vowels and consonants
A) There are 34 ways to split up the 5 consonants around the 3 vowels.
For each of those ways, the consonants can be permuted in 5!/2! ways.
34*5!/2! = 2040
5000 = CCCCCVVV
4100
4010
4001
3200
3020
3002
3110
3101 = CCCVCVVC
3011
2300
2030
2003
2210
2201
2120
2102
2012
2021
1400
1040
1004
1310
1301
1220
1202
1022
1211
1121
1112
1040
1004
1031
1013
See what you can do for B and C.
To find the number of ways letters of the word "MULTIPLE" can be arranged, we need to consider the different cases specified in the question:
A) Without Changing the Order of Vowels:
In this case, we have 3 vowels in the word: U, I, and E. To find the number of arrangements of just the vowels, we can consider them as individual elements to arrange. Since we have 3 vowels, we have 3! = 3 x 2 x 1 = 6 ways to arrange the vowels. The remaining consonants, M, L, T, P, can be arranged in 5! = 5 x 4 x 3 x 2 x 1 = 120 ways. Therefore, the total number of arrangements without changing the order of vowels is 6 x 120 = 720.
B) Keeping the Position of Each Vowel Fixed:
In this case, we keep the vowels U, I, and E at their positions. So, we only have to arrange the 4 consonants: M, L, T, and P. Similar to the previous case, we have 4! = 4 x 3 x 2 x 1 = 24 ways to arrange the consonants.
C) Without Changing the Relative Order/Position of Vowels and Consonants:
To find the number of arrangements without changing the relative order of vowels and consonants, we can treat the vowels and consonants as separate groups. The vowels U, I, and E can be arranged among themselves in 3! = 6 ways, and the consonants M, L, T, and P can be arranged among themselves in 4! = 24 ways. Since the vowels and consonants are treated as separate groups, we multiply the number of arrangements for each group: 6 x 24 = 144.
So, to summarize:
A) Without changing the order of vowels: 720 ways
B) Keeping the position of each vowel fixed: 24 ways
C) Without changing the relative order/position of vowels and consonants: 144 ways