# Aptitude - Permutation and Combination - Discussion

Discussion Forum : Permutation and Combination - General Questions (Q.No. 13)
13.
In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?
10080
4989600
120960
None of these
Explanation:

In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.

Thus, we have MTHMTCS (AEAI).

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.

 Number of ways of arranging these letters = 8! = 10080. (2!)(2!)

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

 Number of ways of arranging these letters = 4! = 12. 2!

Required number of words = (10080 x 12) = 120960.

Discussion:
35 comments Page 1 of 4.

Sourabh sharma said:   4 years ago
Sir how to find the total arrangements of word ;mathematics, of both A and both M together
AA. MM

Prashanth D said:   5 years ago
GIVEN WORD = M A T H E M A T I C S = TOTAL = 11.

Solution:-

VOWEL FROM THE WORD = A A E I -------------------------> (1)
REMAINING LETTERS IN THE WORD = M T H M T C S ------->(2)

CONSIDER [ A A E I ] M T H M T C S = 1 + 7 = 8! divided by 2! 2! [becus we have M M and T T].

FROM ------->A A E I we get 4! and divided 2! [becus we have two A A].

8! 4! DIVIDED BY 2! 2! 2! =======> 120960.
(3)

@Popra.

In the arrangement of letters of word PENCIL, P and C next to each other means P and C always come together and only one possible arrangement of P and C letter that is PC, CP is wrong. Now think of available letters E, N, I, L, [PC].

So possible arrangement = 5! => 120.

Popra Tetseo said:   6 years ago
Anyone can please solve this for me? In how many ways PENCIL be arranged so that P and C are next to each other?
(1)

Rakesh said:   6 years ago
But here are 6 odd spaces and 4 vowels shouldn't we consider that? Please explain to me.

Chandan said:   6 years ago
In the above solution as suggested in the answer section, we are just taking the 4 vowels set at last, but its not said in the question that the vowels set to come together and at last of the word. The words can also be formed as M (AEAI) THMTCS or more which is not considered.

Gouthami said:   6 years ago

Shree said:   6 years ago
Thanks all for the clear explanations.

Krishan Senarath said:   6 years ago
If four countries are contesting for 5 cups in a competition, how many results can be there?

Rachael said:   6 years ago
How can we know that how to separate the vowels?