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without the Ts, there are 5 other distinct letters: A I U D E these 5 letters can arrange themselves in 5! or 120 ways
now, to ensure that no two Ts are together, there must be at least one letter between any two Ts . If we represent each of the other letters as X, then the arrangement can be _X_X_X_X_X_ here the three Ts can take any 3 of the 6 vacant positions, and this will also ensure no two Ts are together
number of ways three positions can be taken out of the six vacant places = 6C3 = 20 hence, total number of ways = 120 x 20 = 2400 (option B)
Re: In how many ways can letters the word ATTITUDE be rearranged such that
[#permalink]
08 May 2020, 10:05
1
Kudos
In how many ways can letters the word ATTITUDE be rearranged such that no two Ts are adjacent to each other?
A. 1800 B. 2400 C. 3600 D. 4320 E. 6720
Total arrangements = 8!/3! = 6720 ways
Case I) All the T's are together - 6! = 720 ways
Case II) exactly 2 T's are together
For 1st position - 1 T out of 3 -> 3 ways For 2nd position - 1 T out of the remaining 2 -> 2 ways For 3rd position - No T -> 5 ways For 4th position - Any remaining alphabets -> 5 ways For 5th position - 4 ways For 6th position - 3 ways For 7th position - 2 ways For 8th position - 1 ways
=> 3*2*5*5*4*3*2*1 = 3600
Therefore total combinations of no T's together = 6720-720-3600 = 2400 ways
Re: In how many ways can letters the word ATTITUDE be rearranged such that
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10 Dec 2020, 09:40
1
Kudos
Expert Reply
Top Contributor
Bunuel wrote:
In how many ways can letters the word ATTITUDE be rearranged such that no two Ts are adjacent to each other?
A. 1800 B. 2400 C. 3600 D. 4320 E. 6720
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.
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