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
To solve each part of the question, we need to understand the principles of permutations and combinations.
A) To find the number of ways to arrange the letters of the word "MULTIPLE" without changing the order of vowels, we need to consider only the consonants. In this case, the consonants are M, L, T, P, and L (as the letter E is a vowel). There are 5 consonants, and the number of ways to arrange them can be calculated as 5!, which equals 5 x 4 x 3 x 2 x 1 = 120. Therefore, there are 120 ways to arrange the letters without changing the order of the vowels.
B) To find the number of ways to arrange the letters of the word "MULTIPLE" while keeping the position of each vowel fixed, we need to consider only the consonants' arrangements. Using the same reasoning as before, we have 5 consonants that need to be arranged. The number of ways to arrange them can be calculated as 5!, which again equals 120. Therefore, there are 120 ways to arrange the letters while keeping the position of each vowel fixed.
C) To find the number of ways to arrange the letters of the word "MULTIPLE" without changing the relative order/position of vowels and consonants, we can consider treating the vowels (U, I, E) and consonants (M, L, T, P, L) as separate groups. We then count the number of ways to arrange the groups and multiply them together. For the vowels group (UIE), there are 3! ways to arrange them, which equals 3 x 2 x 1 = 6. For the consonants group (MLTPLL), there are 6! ways to arrange them, which equals 6 x 5 x 4 x 3 x 2 x 1 = 720. Therefore, there are 6 x 720 = 4320 ways to arrange the letters without changing the relative order/position of vowels and consonants.