To calculate how many passcodes of 3 to 6 digits can be formed using the digits from your number $251414253$, we proceed step by step.

Step 1: Identify the available digits

Your number consists of the following digits: $2, 5, 1, 4, 1, 4, 2, 5, 3$ So, the available digits are: $1, 1, 2, 2, 3, 4, 4, 5, 5$. These digits can be repeated.

Step 2: Possible passcode lengths

The passcodes can have lengths of 3, 4, 5, or 6 digits.

Step 3: Total number of distinct digits

The distinct digits in your number are $\{1, 2, 3, 4, 5\}$. There are 5 distinct digits.

Step 4: Calculate the number of passcodes

Since the digits can repeat, we use the rule of multiplication. For each position in the passcode, we have 5 choices (the distinct digits). Therefore, the number of passcodes of length $n$ is $5^n$.

1. Passcodes of length 3: $5^3 = 125$
2. Passcodes of length 4: $5^4 = 625$
3. Passcodes of length 5: $5^5 = 3,125$
4. Passcodes of length 6: $5^6 = 15,625$

Step 5: Sum of all passcodes

Now, add the total number of passcodes for each length: $125 + 625 + 3,125 + 15,625 = 19,500$

Thus, you can form 19,500 passcodes of lengths between 3 and 6 digits using the digits of your number.

Would you like more details or have any questions?

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Here are 5 related questions for you to explore:

1. What if the passcodes could only be formed using non-repeated digits?
2. How many passcodes can be formed if the passcode length is fixed at