Answer:
In how many ways can the lleter of the word 'Algebra be arranged without changing the relative order of the vowels and consonants?
Step-by-step explanation:
Hint: Here the given question is based on the concept of permutation. Here we have the word ‘ALGEBRA’ where we are not changing the order of the vowels and consonants. Since it is an arrangement, we use permutation concepts and we determine the solution for the question.
Complete step by step solution:
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.
Now consider the given question, here we have the word ‘ALGEBRA’ we have to find the number of ways where the word is arranged such that the vowels and consonants are arranged without changing its order.
In the word ‘ALGEBRA’ there are 3 vowels namely, A, E, A. and 4 constants.
The number of arrangements of the vowels A, E, A where A is repeated once.
Therefore the arrangements of vowels is 3!2!
on simplifying we have
⇒3×2×12×1
On cancelling the terms which is common in both numerator and denominator and then we have
⇒3ways
Therefore in 3 ways the vowels are arranged in the word ALGEBRA
The number of arrangements of the consonants L, G, B, R, where none of the letters are repeated.
Therefore the arrangements of consonants is 4!
on simplifying we have
⇒4×3×2×1
On multiplying the terms
⇒24ways
Therefore in 24 ways the consonants are arranged in the word ALGEBRA.
Therefore the number of arrangements in the word ‘ALGEBRA’ without changing the order of vowels and consonants is 3×24
On simplifying it we have
⇒72ways
Therefore in 72 ways the letters of the word ‘ALGEBRA’ are arranged without changing the order of vowels and consonants.
So, the correct answer is “72 ways”.
Note: Here in this question there is no direct method where we apply to get the solution. So first we find the number of ways the letters are arranged in the word. and then we find the number of ways the letters are arranged in the word such that the two A’s are together. Here we must know the permutation concept and the formula n!=n×n−1×...×2×1
Thanks
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
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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In how many ways can the letters of the word 'ALGEBRA'be arranged without changing the relative order of the vowels and consonants ?
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Solution
There are 4 consonants in the word 'ALGEBRA'.
The consonants are L, G, B, R. they 4 letters.
So, the number of ways to arrange these consonants = 4!
There are 3
vowels in the given word of which 2 are A'S
The vowels can be arranged among themselves in
3!2! ways.
Hence, the required
number of arrangements
=4!×3!2
= 4×3×2×3×22
= 72
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