Aptitude - Permutation and Combination - Discussion

I am not getting this answer please sir give me a clear explanation.

@ALL.

We should notice that in the word "mathematics'", m and t occurred twice 8!/2!2!.

In the word MATHEMATICS, We treat the vowel as one letter. Thus we have MTHMTCS (AEAI). Now we have t arrange 7 letters, out of which M and T occur twice. So no of arranging these letters = 7!/2!*2! = 1260.

Now, AEAI has 4 letters in which A occurs twice.

So, no of ways of arranging these letters = 4!/2!=12.
Required no of words= 1260*12=15120.
So answer is none of these.

MATHEMATICS.

# The consonants MTHMTCS are considered as 7 letters.

# The four vowels AEAI are considered as one letter since they are supposed to be arranged all together.

# The 7 consonant + 1 group of vowels= 8 letters.

# If there are no repeated letters in these 8 letters, the total ways of arrangements would be 8 != 40320.

# But in the 8 letters we set up above we know M and T are happened to occur twice. Since a letter repeated twice has two arrangements M and T add up to have 4 repeated arrangements. These 4 repeated arrangements will divide the total 40320 arrangements in to 4. Hence, 40320÷4= 10080.

# The AEAI letter will form (4!) 24 arrangements if there is no letter repeated. But A is repeated twice which divide the 24 arrangements by 2. Hence, we do have 12. arrangements.

# Finally the total number of ways to arrange is፡.

10080*12=120960 ways.

If vowels never come together then?

How can we know that how to separate the vowels?

If four countries are contesting for 5 cups in a competition, how many results can be there?

Thanks all for the clear explanations.

My answer is 11!/2!2!2!. Is it is wrong, please explain the answer I have a dought.

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.