Sure! Let's assume that the lottery consists of choosing three digits from 0-9, and the order of the digits matters.

The total number of possible combinations is the number of ways to choose 3 digits from 10, which is given by the formula:

10 * 9 * 8 = 720

This means that there are 720 possible winning combinations of 3 digits.

To match the first three digits, there is only one winning combination, for example 1-2-3. Therefore, the probability of matching the first three digits is:

1/720 = 0.0014 or approximately 0.001%

Now, let's consider a scenario where the order of the digits does not matter and each winning digit is unique. In this case, the total number of possible combinations is given by the formula for combinations:

10 choose 3 = 10!/(3!7!) = 120

This means that there are 120 possible winning combinations of 3 digits.

To match the first three digits, there are 3 possible winning digits (for example, if the winning number is 1-2-3, then any of the digits 1, 2, or 3 can be the first digit). Therefore, the probability of matching the first three digits is:

3/120 = 0.025 or approximately 0.006%

I hope this example helps to clarify the mathematical proof behind the probability calculation.
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