SOLUTION: In how many ways you can arrange the letters of the word THINK so that the T and the K are separated by at least one letter? Thank you,,

Question 1115915: In how many ways you can arrange the letters of the word THINK so that the T and the K are separated by at least one letter?
Thank you,,
Answer by greenestamps(13208) (Show Source): You can put this solution on YOUR website!

It useful mental exercise to consider different ways of counting the number of arrangements, seeing that the different ways give the same answer. I immediately see three different ways to do the counting.

(1) List the pairs of positions the T and K can be in; for each of those pairs, the remaining 3 letters can be arranged in 3!=6 different ways.
1 and 3; 1 and 4; 1 and 5; 2 and 4; 2 and 5; and 3 and 5; and the same pairs in the opposite order.

Answer: 12 pairs of positions for the T and K; 6 arrangements of INK for each, so 12*6 = 72 arrangements.

(2) For each possible position of the letter T, determine the number of possible positions for letter K; again for each of those, the remaining letters can be arranged in 6 different ways.
T first --> 3 possible positions for K
T second --> 2 possible positions for K
T third --> 2 possible positions for K
T fourth --> 2 possible positions for K
T fifth --> 3 possible positions for K

That makes a total of 12 different pairs of positions for T and K.

And again the answer is 12*6 = 72 arrangements.

(3) Subtract the number of arrangements in which the T and K are NOT separated by at least one letter from the total number of arrangements of the 5 letters.

The total number of arrangements is 5!=120.

There are 4 pairs of positions in which the T and K can be together: 1 and 2, 2 and 3, 3 and 4, and 4 and 5.
In each of those pairs, the T and K can be in either of two orders.
And again the remaining three letters can be arranged in 6 different ways.

Total number of arrangements in which T and K are together: 4*2*6 = 48

Number of arrangements in which T and K are separated by at least one letter: 120-48 = 72.
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