Question 1115915
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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.<br>
(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.<br>
Answer: 12 pairs of positions for the T and K; 6 arrangements of INK for each, so 12*6 = 72 arrangements.<br>
(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<br>
That makes a total of 12 different pairs of positions for T and K.<br>
And again the answer is 12*6 = 72 arrangements.<br>
(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.<br>
The total number of arrangements is 5!=120.<br>
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.<br>
Total number of arrangements in which T and K are together: 4*2*6 = 48<br>
Number of arrangements in which T and K are separated by at least one letter: 120-48 = 72.