How many ways, can the letters of the word 'QUANTITATIVE' be arranged…

The correct answer is Option (4) → 907200

1. Count the Frequency of Each Letter

First, let's identify the letters in 'QUANTITATIVE':

• Q: 1
• U: 1
• A: 2
• N: 1
• T: 3
• I: 2
• V: 1
• E: 1
• Total Letters: 12

2. Group the Required Letters

Since all 'T's must be together, we treat the three 'T's as a single block or unit: {TTT}.

Now, we count the total number of units to be arranged:

• The block {TTT}: 1 unit
• Remaining letters: Q(1), U(1), A(2), N(1), I(2), V(1), E(1) = 9 units
• Total Units to arrange: $1 + 9 = 10$ units

3. Calculate the Permutations

The number of ways to arrange these 10 units, accounting for the repeating letters (A appears 2 times and I appears 2 times), is given by the formula:

$\text{Arrangements} = \frac{n!}{p! \cdot q!}$

Where $n$ is the total units, and $p, q$ are the frequencies of repeating units.

$\text{Arrangements} = \frac{10!}{2! \cdot 2!}$

$\text{Arrangements} = \frac{3,628,800}{2 \cdot 2}$

$\text{Arrangements} = \frac{3,628,800}{4} = 907,200$