# In how many ways can the letters of the word ‘ALGEBRA’ be arranged without changing the relative order of the vowels and consonants?

Given, the word ‘ALGEBRA’. It has 7 letters of which 4 of them are consonants (L, G, B, R) and 3 of them are vowels (A, E, A) repeating the vowel A twice.

We have to find a number of words that can be formed without changing the relative order of vowels and consonants, i.e. if a vowel comes before a consonant in word ALGEBRA, it has to be in the same order in all possible words. For example- ‘ELGABRA’ and ‘AGLEBRA’ are such two words.

Since we know, Permutation of n objects taking r at a time is nPr,and permutation of n objects taking all at a time is n!

And, we also know Permutation of n objects taking all at a time having p objects of the same type, q objects of another type, r objects of another type is . i.e. the, number of repeated objects of samethe type are in denominator multiplication with factorial.

The consonants in their positions can be arranged in 4! = 24 number of ways, since there is no repeating letter in consonants set (L, G, B, R).

Similarly, the vowels in their positions can be arranged in 3! / 2! = 3 number of ways, since there is one repeated letter in vowel set (A, E, A), i.e. the letter A repeating twice.

Now, total number of arrangements where relative position of consonants and vowels are not changed will be exactly equal to number of ways we can select one to one element from consonant set to vowel set using Multiplication principle (If an event A can occur in m different ways and another event B can occur in n different ways then a total number of ways of simultaneous occurrence of both events in definite order is m x n.)

i.e. Total arrangements = 24 x 3

= 72

Hence, a total number of ways the word ‘ALGEBRA’ be arranged such that the relative position of vowels and consonants are unchanged is equaled to 72.

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