A company randomly assigns work identification numbers to their employees. Each
number consists of three digits from one to nine. If the digits cannot be repeated, find the
probability for a randomly generated number to be 315.
The identification numbers have three digits, so each of these digits can be a
number from one to nine. So the very first digit can be any number from one to nine, so there’s nine options, nine different numbers
to choose from.
The second digit, however, yes it is any number from one through nine, but the digits
cannot be repeated once they’ve been used. So if there’s nine options for the first digit, there’s
only gonna be eight options for the second digit, because one of the numbers from one through nine was used for the first digit.
And that number cannot be repeated, so it is no longer an option for the second digit.
This means, for the third digit, since two of the numbers from one through nine
have already been used, now there’s only seven numbers left to choose from.
In order to find the total number of possible identification numbers, we would
need to multiply each of these digits together. So nine times eight times seven is 504. There’s 504 total ID numbers that could
be used. So the probability for a randomly generated number to be 315
would be one out of 504, because, out of all of the total numbers that could be used, there’s only one
number that’s 315, so one out of the 504.
Just as a side note, there is another way to do this question, using permutations. Generally, the number of permutations of 𝑟 objects chosen from 𝑛 is given by 𝑛
factorial divided by 𝑛 minus 𝑟 factorial.
So in this case, 𝑛 would be nine and 𝑟 would be three. So we have nine factorial over six factorial.
Nine factorial is nine times eight times seven times six times five times four
times three times two times one. Six factorial is six times five times four times three times two times one.
So here we can see the six, five, four, three, two, and ones cancel, so we have nine
times eight times seven,
which is equal to the 504 like we had found.
So when finding the
probability of a randomly generated number to be 315, that would be one out of 504.