I form a 3-digit number at random from the digits 1-9. What is the probability that my number has no repeated digits? All three digits the same? Exactly one repeated digit?

please?

number of cases with no repeating digits

= 9x8x7 = 504
number of cases where digits repeat
= 9x9x9 = 729
so prob(no repeating digits) = 504/729 = 56/81

all 3 digits repeat, could be 111 , 222, .. 999
or 9 of them
prob = 9/729 = 1/81

I am not clear what you mean by "exactly one repeated digit"

one digit is repeated twice

To determine the probability of different outcomes, we need to calculate the total number of possible outcomes and then count the number of favorable outcomes for each situation.

1. Probability of no repeated digits:
To find the number of 3-digit numbers with no repeated digits, we can use the concept of permutations because the order of the digits matters.
Since we have 9 digits to choose from for the first position, 8 digits left for the second position, and 7 remaining digits for the third position, the total number of possible outcomes is 9 * 8 * 7 = 504.
Since there are 9 possible digits for the first position, and then only 8 and 7 for the second and third positions respectively (without replacement), the number of favorable outcomes is 9 * 8 * 7 = 504.

Therefore, the probability of forming a 3-digit number with no repeated digits is 504 / 504 = 1 (or 100%).

2. Probability of all three digits the same:
To find the number of 3-digit numbers with all three digits the same, we only have one digit to choose from for all three positions.
Therefore, the number of possible outcomes is 1 * 1 * 1 = 1 (there's only one way to form this number), and the number of favorable outcomes is likewise 1.

So, the probability of forming a 3-digit number with all three digits the same is 1 / 1 = 1 (or 100%).

3. Probability of exactly one repeated digit:
To find the number of 3-digit numbers with exactly one repeated digit, we can determine the favorable outcomes by choosing a digit to be repeated and selecting two unique digits for the other positions.
There are 9 choices for the digit to be repeated, and for the second position, there are 8 choices (since it must be different from the repeated digit), and for the third position, there are 7 choices (since it should also be different from both previous digits).

Therefore, the number of favorable outcomes is 9 * 8 * 7 = 504.

The total number of possible outcomes is still 9 * 8 * 7 = 504 (as explained in part 1).

Hence, the probability of forming a 3-digit number with exactly one repeated digit is 504 / 504 = 1 (or 100%).

In summary, the probabilities are as follows:
- Probability of no repeated digits: 1 (or 100%)
- Probability of all three digits the same: 1 (or 100%)
- Probability of exactly one repeated digit: 1 (or 100%)