IGDTUW 2025 — Mathematics PYQ

Step 1: Identify the letters in 'MATHEMATICS'

The word 'MATHEMATICS' has 11 letters in total:

• Vowels: A, E, A, I (4 vowels)

• Consonants: M, T, H, M, T, C, S (7 consonants)

Note the repetitions: M appears 2 times, T appears 2 times, and A appears 2 times.

Step 2: Group the vowels together

Since the vowels (A, A, E, I) must always come together, we treat them as a single unit or block.

Now, we have:

• The 7 consonants + 1 vowel-block = 8 items to arrange.

Step 3: Arrange the items

Among the 7 consonants (M, M, T, T, H, C, S), there are duplicates: 2 M's and 2 T's.

The number of ways to arrange these 8 items is:

$$\frac{8!}{2!\times 2!}=\frac{40320}{4}=10080$$

Step 4: Arrange the vowels within the block

Within the vowel block (A, A, E, I), there are 4 letters with the vowel A repeating 2 times.

The number of ways to arrange the vowels among themselves is:

$$\frac{4!}{2!}=\frac{24}{2}=12$$

Step 5: Calculate the total arrangements

To find the final answer, multiply the arrangements of the items by the internal arrangements of the vowel block:

$$\text{Total ways}=10080\times 12=120960$$

Conclusion:

The total number of ways the letters can be arranged is 120960. Comparing this to the given options, the correct choice is (c).