How many ways, can the letters of the word 'QUANTITATIVE' be arranged, so that all T are together?

The word QUANTITATIVE has 12 letters: Q-U-A-N-T-I-T-A-T-I-V-E

The letters that repeat are:

• A appears 2 times
• T appears 3 times
• I appears 2 times
• All other letters (Q, U, N, V, E) appear once

When all T's must be together, treat the three T's as one single block [TTT].

After grouping the T's together:

Original count: 12 letters

New count: 12 - 3 + 1 = 10 units

The 10 units to arrange are: [TTT], Q, U, A, N, A, I, V, I, E

Among these 10 units:

• A repeats 2 times
• I repeats 2 times
• The [TTT] block counts as 1 unit (no repetition)

Number of arrangements $= \dfrac{10!}{2! \times 2!}$

$10! = 3,628,800$

$2! = 2$

Number of ways $= \dfrac{3,628,800}{2 \times 2}$

$= \dfrac{3,628,800}{4}$

$= 907,200$

Therefore, the letters of QUANTITATIVE can be arranged in 907,200 ways with all T's together.