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Thread: Multiplication Principle

  1. #1
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    Multiplication Principle

    Each student at a school is given a 6 digit code (like 123876 or 001233)

    (b) How many codes read the same forward as backward

    I think I understand this one.. For a 6 digit number to be same in forward/reverse, I would have to count the possibilities of the first 3 digits? I'm not sure why this is or if it is even right.

    (d) How many codes contain at least one even digit

    If each digit can hold 0-9, and there are only 5 even numbers in that interval... would it be 5^6?



    (c) How many numbers contain only odd digits?

    I know this one is 5^6, but I'm curious if this is the same concept as (d) with different wording?
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  2. #2
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    Re: Multiplication Principle

    b) pick any three digit code, there are 1000 of them, these are the first 3 digits, reverse them for the last three digits

    d) easiest to compute the number of codes w/no even digits and subtract it from the number of codes.

    # of codes with no even digits is $5^6 = 15625$

    # of 6 digit codes is $10^6 = 1000000$

    # of 6 digit codes w/at least 1 even digit = $1000000 - 15625 = 984375$

    c) is not the same as (d). However, the codes in (d) plus the codes in (c) make up all the possible codes.
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    Re: Multiplication Principle

    Quote Originally Posted by MrJank View Post
    Each student at a school is given a 6 digit code (like 123876 or 001233)
    (b) How many codes read the same forward as backward
    I think I understand this one.. For a 6 digit number to be same in forward/reverse, I would have to count the possibilities of the first 3 digits? I'm not sure why this is or if it is even right.
    (d) How many codes contain at least one even digit
    If each digit can hold 0-9, and there are only 5 even numbers in that interval... would it be 5^6?
    (c) How many numbers contain only odd digits?
    There are $10^6$ possible 6-digits codes.

    b) There are $10^3$ 3-digits codes So there are that many 6-digit codes which read either way. WHY? HOW?

    Do c) next! odd digits, $\{1,~3,~5,~7,~9\}$ five in all. Thus there $5^6$ codes with only odd digits.

    d) Is the answer here $10^6-5^6~?$
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