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A sequence of letters of the form $abcba$, where the expression is unchanged upon reversing order, is an example of a palindrome (of five letters).

(a) if a letter may appear more than twice, how many palindromes of five letters are there? of six letters

(b) Repeat part (a) under the condition that no letter appears more than twice

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So, what are yourr thoughts on the questions? How far can you get? what do you know about permutations? Why are you interested in this particular problem? If it's homework, that's OK, but please edit in the "homework" tag, so we have a better idea of the kind of answer to write. –  Gerry Myerson Feb 9 '13 at 5:12
    
for part (a) : I use this way, There are four possibilities form of the palindrome, they are aaaaa ; baaab; ababa; aabaa; the possibilities of the first form is 26, the possibilities of the second form is 26 x 25 the possibilities of the third form is also 26 x 25 the possibilities of the fourth form is 26 x 25 so, the number of palindromes of five letters with a letter may appear more than twice is 26 + (3 x 36 x 25) is this correct? –  chihiroasleaf Feb 9 '13 at 5:13
    
it's not homework.., I'm preparing for the test on the next week... –  chihiroasleaf Feb 9 '13 at 5:14
    
I think you've left out abcba. –  Gerry Myerson Feb 9 '13 at 5:16
    
hhmm..., so I should add (26 x 25 x 24) ? and it will be 6 + (3 x 36 x 25) + (26 x 25 x 24) right? –  chihiroasleaf Feb 9 '13 at 5:21
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1 Answer

Hint: for a) the first half (round up-why?) determines the rest. For b) this is a new restriction on the first half. What about the middle letter?

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what do you mean by round-up? I've tried to solve half of the part (a). I write it on the comment above.., is it correct? –  chihiroasleaf Feb 9 '13 at 5:17
    
@chihiroasleaf: I meant if you have a word of five letters, it takes three to determine the rest. If you have a word of six, it still takes three. Gerry Myerson's comment is on point. –  Ross Millikan Feb 9 '13 at 5:19
    
@chihiroasleaf: for b, if a letter appears twice in the determining part (the first three) it will appear three or four times in the complete word, so you need the first three letters distinct. –  Ross Millikan Feb 9 '13 at 16:40
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