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A fair, six-sided die is rolled eight times, to form an eight-digit number. What is the probability that the resulting number is a multiple of 8? Express your answer as a common fraction.

  • math -

    Total number of numbers using 1-6
    = 6^8 = 1679616

    For any number to be divisible by 8, its last 3 digits must be divisible by 8
    but these last two digits must contain only the digits from 1 to 6, with no zeros.

    possible cases are:
    112 136 144 152
    216 224 232 256 264
    312 336 344 352
    416 424 432 456 464
    512 536 544 552
    616 624 632 656 664

    Hoping that I didn't miss any, I count 27

    so the front 5 numbers could be anything
    there are 6^5 or 7776 different front numbers, each of those could have 27 different last three numbers
    so there are 7776x27 or 209952 which are divisible by 8

    prob that a number is a multiple of 8
    = 209952/1679616
    26244/209952
    = 6561/52488
    = 1/8

    mmmmh, very interesting

  • math -

    That is right

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