You pick 6 digits (0-9) at random without replacement, and write them in the order picked.

What is the probability that you have written the first 6 digits of your phone number? Assume there are no repeats of digits in your phone number.

Give your answer as a fraction.

To solve this problem, let's break it down step-by-step.

### Step 1: Determine the total number of possible 6-digit sequences
When picking 6 digits randomly without replacement from the digits 0-9, we are choosing 6 digits out of 10 and arranging them in a specific order.

The number of ways to choose 6 digits out of 10 is given by the binomial coefficient \(\binom{10}{6}\). However, since we also care about the order in which these digits are picked, we multiply by \(6!\), the number of ways to arrange 6 digits.

So, the total number of possible 6-digit sequences is:

\[
\text{Total number of sequences} = \frac{10!}{(10-6)!} = \frac{10!}{4!} = 10 \times 9 \times 8 \times 7 \times 6 \times 5
\]

### Step 2: Determine the number of favorable sequences
Now, there is exactly 1 sequence that matches the first 6 digits of your phone number. Since we need an exact match, there is only 1 such favorable outcome.

### Step 3: Calculate the probability
The probability of a specific sequence matching the first 6 digits of your phone number is the ratio of the number of favorable sequences to the total number of sequences.

\[
\text{Probability} = \frac{\text{Number of favorable sequences}}{\text{Total number of sequences}} = \frac{1}{10 \times 9 \times 8 \times 7 \times 6 \times 5}
\]

Now let's compute the value:

\[
10 \times 9 \times 8 \times 7 \times 6 \times 5 = 151200
\]

Thus, the probability is:

\[
\frac{1}{151200}
\]

### Final Answer
The probability that the 6 digits chosen randomly match the first 6 digits of your phone number is:

\[
\boxed{\frac{1}{151200}}
\]

Would you like further details or have any questions about this solution?

Here are some related questions to consider:

1. What is the probability if you picked 7 digits instead of 6?
2. How would the probability change if the digits could be repeated?
3. What is the probability that the first and last digits match two specific numbers in your phone number?
4. How many ways are there to arrange 10 digits without repetition?
5. What is the probability of getting any particular 4-digit sequence in a random 6-digit sequence?

**Tip:** When calculating probabilities with sequences, consider the total number of possible sequences and the number of favorable outcomes.

You pick 6 digits (0-9) at random without replacement, and write them in the order picked. What is the probability that you have written the first 6 digits of your phone number? Assume there are no repeats of digits in your phone number. Give your answer as a fraction.

Learn how to calculate the probability of matching the first 6 digits of your phone number when picking 6 digits randomly from 0-9 without replacement. This problem explores combinatorics, permutations, and probability concepts suitable for high school students.

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