Can someone help? Very confused. . .
John Sununus was once the governor of New Hampshire, and his name reminds one of the authors of a palindrome (a words which is spelt the same way forwards as backwards, such as SUNUNUS).
How many seven-letter palindromes (not necessarily real words) begin with the letter S and contain at most three letter?
Thanks for any helpful replies!
See response to previous post:
http://www.jiskha.com/display.cgi?id=1302642083
To find the number of seven-letter palindromes that begin with the letter S and contain at most three distinct letters, you can break down the problem into smaller, manageable steps.
Step 1: Identify the middle letter of the palindrome.
Since palindromes read the same forwards and backwards, the middle letter must be the same as the first letter. Therefore, we can conclude that the first letter will be S.
Step 2: Determine the possible letters for the remaining five positions.
Since the middle letter is fixed as S, there are only two possible distinct letters remaining. Let's call these letters X and Y. We have three cases to consider, depending on the number of distinct letters used.
Case 1: Using only one distinct letter.
In this case, all five remaining letters will be the same. Since we have two possible letters (X and Y), we have two options for this case.
Case 2: Using two distinct letters, with three of one letter and two of the other.
In this case, we need to choose two distinct letters from X and Y. Since the order does not matter, we treat it as a combination problem. We have two options for the first letter and one option for the second. Therefore, there are a total of two options for this case.
Case 3: Using two distinct letters, with two of each letter.
In this case, we again need to choose two distinct letters from X and Y. Since the order does not matter, we treat it as a combination problem. We have two options for the first letter and one option for the second. Therefore, there are a total of two options for this case.
Step 3: Calculate the total number of palindromes.
To find the total number of palindromes, we sum up the number of palindromes in each case:
Case 1: 2 options
Case 2: 2 options
Case 3: 2 options
Therefore, the total number of seven-letter palindromes that begin with the letter S and contain at most three distinct letters is 2 + 2 + 2 = 6.
So, there are six such palindromes.