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.
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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 <--- Very suspicious
BOB is right.
To determine the probability that the resulting number is a multiple of 8, we need to consider a few key principles:
1. A number is divisible by 8 if the last three digits of that number (when read from right to left) form a multiple of 8. In other words, we only need to consider the last three digits of the eight-digit number to determine its divisibility by 8.
2. For a number to be divisible by 8, it must be even and divisible by 4. Thus, we need to count the number of possible three-digit numbers that are divisible by 8.
To count the number of three-digit multiples of 8, we can use a counting strategy:
1. Start by finding the smallest three-digit multiple of 8, which is 104.
2. The second smallest three-digit multiple of 8 is 112 (104 + 8).
3. Observe that each subsequent three-digit multiple of 8 is 8 units larger than the previous one.
Using this pattern, we can conclude that the possible three-digit multiples of 8 range from 104 to 992, with increments of 8.
Now, let's calculate the total number of outcomes for rolling an eight-sided die eight times.
Since each roll produces one of six equally likely outcomes (numbers 1 to 6), the total number of outcomes for eight rolls of the die is 6^8 = 16777216.
Next, let's determine how many ways we can arrange eight digits to form the eight-digit number. Since the order of the digits doesn't matter for our purpose, we need to calculate the number of combinations.
We can do this using the concept of combinations. The number of ways to choose three elements from a set of eight is given by the binomial coefficient C(8,3), which can be calculated as:
C(8,3) = 8! / (3!(8-3)!) = 56
Now, the probability that the resulting number is a multiple of 8 is the ratio of the favorable outcomes (the number of three-digit multiples of 8, which is 56) to the total outcomes (16,777,216):
Probability = favorable outcomes / total outcomes
Probability = 56 / 16,777,216
Simplifying the fraction, we have:
Probability = 1 / 300,425
Thus, the probability that the resulting number is a multiple of 8 is 1/300,425, which is the answer expressed as a common fraction.