Answer

Verified

450k+ views

Hint: Find the positions of vowels and consonants in the word PATALIPUTRA where vowels can be A, E, I, O, U and the rest of the English alphabets are consonants. Here we have to arrange the letters according to given conditions in the problem. For arrangement we use the permutation formula.

Complete step-by-step answer:

As we can see that in the word PATALIPUTRA there are 5 vowels and 6 consonants.

And the positions of vowels are 2nd, 4th, 6th, 8th and 11th.

And the positions of consonants are 1st, 3rd, 5th, 7th, 9th and 10th.

As we can see that vowels are A, A, I, U and A.

So, out of 5 vowels 3 are A (same).

Now, consonants are P, T, L, P, T and R out of 6, 2 are P (same) and 2 are T (same).

So, now as it is given in the question that the relative positions of vowels and consonants should be the same.

So, we can arrange vowels and consonants in their own positions only.

And as we know that if there are n numbers and out of them m are of the same type and r are same of another type.

Then number of ways of arranging n numbers will be \[\dfrac{{{\text{n!}}}}{{{\text{m! r!}}}}\]

So, here 5 vowels in the word PATALIPUTRA can be arranged themselves in \[\dfrac{{{\text{5!}}}}{{{\text{3!}}}}\] as a letter A is repeated thrice.

The remaining 6 consonants can permute among themselves in \[\dfrac{{{\text{6!}}}}{{{\text{2! 2!}}}}\] as two letters . P and T are repeated twice.

Hence, the total number of permutations are \[\dfrac{{{\text{5!}}}}{{{\text{3!}}}}\] * \[\dfrac{{{\text{6!}}}}{{{\text{2! 2!}}}}\] = 3600

So, the correct option will be A.

Note: Whenever we come up with this type of problem then first, we have to find the permutation for arranging vowels and then we will find the permutation of arranging the consonants and then we will multiply them both to get the required total number of ways.

Complete step-by-step answer:

As we can see that in the word PATALIPUTRA there are 5 vowels and 6 consonants.

And the positions of vowels are 2nd, 4th, 6th, 8th and 11th.

And the positions of consonants are 1st, 3rd, 5th, 7th, 9th and 10th.

As we can see that vowels are A, A, I, U and A.

So, out of 5 vowels 3 are A (same).

Now, consonants are P, T, L, P, T and R out of 6, 2 are P (same) and 2 are T (same).

So, now as it is given in the question that the relative positions of vowels and consonants should be the same.

So, we can arrange vowels and consonants in their own positions only.

And as we know that if there are n numbers and out of them m are of the same type and r are same of another type.

Then number of ways of arranging n numbers will be \[\dfrac{{{\text{n!}}}}{{{\text{m! r!}}}}\]

So, here 5 vowels in the word PATALIPUTRA can be arranged themselves in \[\dfrac{{{\text{5!}}}}{{{\text{3!}}}}\] as a letter A is repeated thrice.

The remaining 6 consonants can permute among themselves in \[\dfrac{{{\text{6!}}}}{{{\text{2! 2!}}}}\] as two letters . P and T are repeated twice.

Hence, the total number of permutations are \[\dfrac{{{\text{5!}}}}{{{\text{3!}}}}\] * \[\dfrac{{{\text{6!}}}}{{{\text{2! 2!}}}}\] = 3600

So, the correct option will be A.

Note: Whenever we come up with this type of problem then first, we have to find the permutation for arranging vowels and then we will find the permutation of arranging the consonants and then we will multiply them both to get the required total number of ways.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE