In how many ways can letters the word ATTITUDE be rearranged such that

Asked: In how many ways can letters the word ATTITUDE be rearranged such that no two Ts are adjacent to each other?

A-1
T-3
I-1
U-1
D-1
E-1

Total words formed with letters of ATTITUDE = 8!/3! = 8*7!/6 = 4/3* 7! = 6720

Words in which 2 Ts are adjacent to one another = 7! = 5040

A-1
2T-1
T-1
I-1
U-1
D-1
E-1

But this includes the case when all 3 Ts are together and considers <2T>T & T<2T> as separate cases and are counted twice.

Words in which 3 Ts are adjacent to one another = 6! = 120

Words in which one or more Ts are adjacent to one another = 7! - 6! = 6*6! = 4320

Words in which no Ts are adjacent to one another = 6720 - 4320 = 2400

IMO B

Take the task of seating the letters and break it into stages.

Stage 1: Arrange the non-T letters (A, I, U, D and E)
Since n unique objects can be arranged in n! ways, we can arrange these five letters in 5! ways. In other words, we can complete stage 1 in 120 ways.

Key step: For each arrangement of the 5 non-T letters, add a space on each side of each letter. So, for example, if we add spaces to the arrangement ADIUE, we get: _A_D_I_U_E_

Each of these six spaces represents a possible location for each of the 3 T's. Notice that this configuration ensures that no T's can be adjacent.

Stage 2: Place T's in 3 of the 6 available spaces .
IMPORTANT: Since we're placing 3 identical T's, the order in which select the 3 spaces doesn't matter.
So, we can use combinations.
We can choose 3 of the 6 spaces in 6C3 ways (= 20 ways)
So, we can complete stage 2 in 20 ways.

At this point, we'll throw away the remaining spaces, leaving an arrangement with all 8 letters.

By the Fundamental Counting Principle (FCP), the number of ways to complete the 2 stages (and thus place all 7 letters) = (120)(20) = 2,400 ways.

Answer: B

Cheers,
Brent
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