Question 1149532
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I will assume this question is from the same student that asked the number of words when the letters N and D are separated by EXACTLY two letters....<br>
Then, continuing in the same manner as in my response to that question....<br>
(1) N and D separated by THREE letters: (3)(2)(120) = 720
(2) N and D separated by FOUR letters: (2)(2)(120) = 480
(3) N and D separated by FIVE letters (the maximum): (1)(2)(120) = 240<br>
ANSWER: 1440<br>
Notice we can see that this method of counting the numbers of words with different numbers of letters between N and D is valid by finding the numbers of words for ALL the possible distances between N and D and showing that the sum is 7! =5040.<br>
N and D separated by ONE letter: (5)(2)(120) = 1200
N and D separated by ZERO letters: (6)(2)(120) = 1440<br>
That, together with the above response for this question and the earlier response for the case with 2 letters between N and D, gives us:<br><pre>
0 letters between: 1440
1 letter between:  1200
2 letters between:  960
3 letters between:  720
4 letters between:  480
5 letters between:  240
         --------------
           total   5040</pre><br>