Odd phone numbers have eight digits and none of them start with zero. How many digits are in ascending order? Repetition is not allowed.


To solve this, we need to determine how many 8-digit odd phone numbers exist such that all their digits are in ascending order and without repetition, with none of them starting with zero.

### Steps:

1. **Understand the structure of the phone number:**
   - The number is odd and consists of 8 digits.
   - None of the digits can start with zero.
   - The digits are arranged in ascending order without repetition.
   - The last digit must be odd because the number is odd.

2. **Digits available:**
   - The digits we can use are 1 through 9 (since zero is not allowed at the start, and repetition is not allowed).
   - Among these, the odd digits are 1, 3, 5, 7, and 9.

3. **The last digit must be odd:**
   - Since the number must end in an odd digit, the last digit has to be chosen from {1, 3, 5, 7, 9}.

4. **Choose the remaining 7 digits:**
   - Once the last (odd) digit is chosen, the remaining 7 digits must be selected from the digits smaller than that last digit.
   - For example, if the last digit is 9, we need to pick 7 digits from {1, 2, 3, 4, 5, 6, 7, 8}.
   - If the last digit is 7, we can only pick from {1, 2, 3, 4, 5, 6}.

5. **Summing up possibilities:**
   - For each possible odd last digit, count how many ways we can select 7 digits from the available smaller digits:
     - If the last digit is **9**, we choose 7 digits from {1, 2, 3, 4, 5, 6, 7, 8}. This is \( \binom{8}{7} = 8 \) ways.
     - If the last digit is **7**, we choose 7 digits from {1, 2, 3, 4, 5, 6}. This is \( \binom{6}{7} = 1 \) way.
     - If the last digit is **5**, we choose 7 digits from {1, 2, 3, 4}. This is \( \bin bus 

Odd phone numbers have eight digits and none of them start with zero. How many digits are in ascending order? Repetition is not allowed.

Explore how to calculate the number of 8-digit odd phone numbers where digits are arranged in ascending order and no repetition is allowed. This combinatorics problem highlights key mathematical concepts such as binomial coefficients and odd number properties. Suitable for students in Grades 10-12.

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