Find the number of palindromes of length n over the English...

Key Concepts

Parity in Palindromic Strings

The method for counting palindromes differs based on whether the length of the string is even or odd. For an even-length string, exactly half of the positions are free to choose, whereas for an odd-length string, the middle character can be chosen independently along with the first half, with the remainder determined by symmetry.

Alphabet Size

The size of the alphabet, which is 26 for the English alphabet, determines the number of available choices for each independent character position in the palindromic string. This value is raised to the power of the number of independent choices to obtain the total count of palindromes.

Palindrome

A palindrome is a sequence of characters that reads the same forwards and backwards. This symmetry implies that only a portion of the characters need to be independently chosen, while the rest are fixed by their positions relative to the center of the sequence.

Fundamental Counting Principle

The fundamental counting principle states that if there are multiple independent choices to be made, the total number of outcomes is the product of the number of options available for each choice. This principle is crucial in calculating the number of palindromic strings, where only a subset of the positions are independently selected.

Transcript

00:01 Okay, so in this problem, they are telling us to count the number of palindromes with n letters and length using english letters.

00:09 We're just going to assume they don't really care if they are true words or not because we are just counting ways we can rearrange letters.

00:17 But there's a few things here that we care about.

00:19 First, a palindrome is a word that is read the same forward as it is backwards.

00:25 And second, the word is n letters long.

00:28 So the easiest way to get started with this is just to consider words of, a different length.

00:33 So if we start off with a word that is three letters long and count how many options we have for each letter, the first letter can be anything, any letter in the alphabet.

00:42 But whatever letter it is has to be the same as the last letter.

00:46 So i highlighted them the same.

00:47 Once we've locked in this first letter, we've also locked in this last letter, and now there's only one option for that last letter.

00:54 The middle letter could be anything.

00:56 It could be the same as the first.

00:58 It could be different from the first.

00:59 So we still have 26 options there.

01:00 So our number of options is 26 to the second.

01:05 Four letters is pretty similar, but this time not only did the first and last letter have to be the same, but these two middle numbers have to be the same.

01:13 And so anything i've colored in the same colors, you can kind of see follows that logic of having to be the same so that it mirrors around.

01:20 Again, that means that once we've found one of our letters, it locks in another letter, which means when they have one option for that slot.

01:27 So again, the number of options here is 26 to the second.

01:32 Following this, we have five letters...

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