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Discrete Math

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Hi there I'm confused as to how to start this problem, any help at all would be great thanks!

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?

  • Discrete Math -

    You are looking at a 7 word palindrome, which means that the first four letters will be mirrored about the fourth, for example:

    EXA(M)AXE where mirroring is about the letter M.

    We need therefore all combinations of
    1. a four letter word,
    2. starting with "S",
    3. made up of maximum of 3 letters, meaning that there must be at least one pair of duplicates,
    4. using the 26-letter alphabet.

    There are four positions to fill, of which the first one is already filled with the letter "S".

    There are 26 choices for each of the following 3 positions, giving 26³ words starting with "S".

    Out of the 3³ words, many of them do not have duplicates, i.e. made up of four distinct letters. We need to subtract these words.

    The number of words with distinct characters can be made with
    25 letters for the second position (ecluding "S")
    24 letters for the third position (excluding "S" and the second position) and
    23 letters for the fourth position (excluding the three previous letters).
    for a total of 25*24*23 words.

    So the number of words satisfying the required criteria is:
    26^3-25*24*23 = 3776

  • Discrete Math -

    Oh okay I think I am following. . .

  • Discrete Math -

    So, how many seven-letter palindromes contain at most three different letters one of which is S?

    We would start out with 26^3, but I don't understand how to make sure S will be included as one of the different letters. Any suggestions? Thank you.

  • Discrete Math -

    Since the last three letters of a 7-letter palindrome is identical to the first three, the number of 7-letter palindromes would be the same as the number of distinct 4-letter words.

    In this case, the 4-letter words begin with an "S", and contain only 3 letters, which is the same as requiring that at least 2 of the 4 letters are the same.

    By assuming that the first letter is "S", we only have 3 other letters to fill, hence 26^3.
    Some of these words will have 4 distinct letters, others will contain at least one pair of identical letters.

    The count of distinct letters are eliminated by subtracting the number of ways to make distinct letters:
    1st position: "S"
    2nd position: 26-"S"=25
    3rd position: 26-"S"-2nd=24
    4th position: 26-"S"-2nd-3rd = 23
    So there are 25*24*23 words starting with "S" that are made up of 4 distinct letters.

  • Discrete Math -

    Oh ok so there are 13800 that contain at most three different letters one of which is S.

  • Discrete Math -

    Can you at all help with this?

    Multiple personality disorder (MPD) is a condition in which different personalities exist within one person and at various times control that person’s behavior. In a recent survey of people with MPD, it was reported that “98% had been emotionally abused, 89% had been physically abused, and most had experienced both types of abuse.” Make this statement more precise.

  • Discrete Math -

    Watch out!

    26^3=17576 is the number of ALL 4-letter words starting with "S".

    25*24*23=13800 is the number of 4-letter words starting with "S" that contain exactly 4 distinct letters.

    26^3-25*24*23 = 3776 is the number of 4-letter word starting with "S" that contain 3 or less distinct letters, including "S".

  • Discrete Math - MPD -

    For this problem, do you think you can pinpoint the exact percentage of patients who experienced both, or do you think you can only present a range?

    If it is the exact value, what is it?

    If it is a range, what are the upper and lower limits?

  • Discrete Math -

    Ok now I am confused again with 5:08, it seems the same as the first subpart of the problem. They can't be the same answer, for this one: How many seven-letter palindromes contain at most three different letters one of which is S?

    And for the MPD problem that was really the whole problem. It seems like I have nothing to work with though. . .I'm not sure where to go from here

    You can consider

    n: number of people surveyed.
    E: the set of those who had been emotionally abused.
    P: the set of those who had been physically abused.

  • Discrete Math -

    Basically for the MPD problem I have to make it more precise.

  • Discrete Math -

    For 5:08, the problem was interpreted as 3 letters including "S". The reasoning is we cannot have more than 3 if we exclude the "S" which makes the requirement superfluous.

    If the problem were 3 distinct letters excluding the "S", the 26^3 is the answer.

    For the MPD problem, try drawing possible Venn diagrams using the given information, and you will probably notice some trends or hints.

  • Discrete Math - correction -

    Should read:
    "If the problem were 3 distinct letters or less excluding the "S", the 26^3 is the answer. "

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