total number of 6 digit numbers in which only and all five digits 1,3,5,7 and 9 appear is

To find the total number of 6-digit numbers in which only the digits 1, 3, 5, 7, and 9 appear, we need to consider a few steps.

### Understanding the problem:
- We need **6-digit numbers**.
- The digits allowed are: **1, 3, 5, 7, and 9**.
- All **five digits (1, 3, 5, 7, and 9)** must appear, and they appear exactly **once** each.

Thus, the number must be made up of the digits **1, 3, 5, 7, 9**, plus one of these digits must appear **twice**, because we have 6 digits total, and 5 distinct digits available.

### Step 1: Choose the digit that appears twice
We have 5 choices for the digit that will appear twice (since each of the digits 1, 3, 5, 7, or 9 can be chosen).

### Step 2: Arrange the digits
Once the repeated digit is chosen, we are left with arranging 6 digits, where 5 are distinct and one repeats twice. The number of ways to arrange these 6 digits is given by the formula for permutations of multiset:
\[
\frac{6!}{2!} = \frac{720}{2} = 360
\]
(We divide by \(2!\) because one digit is repeated twice.)

### Step 3: Multiply the choices
Since we have 5 choices for which digit repeats, and for each of these choices there are 360 ways to arrange the digits, the total number of such 6-digit numbers is:
\[
5 \times 360 = 1800
\]

### Final Answer:
There are **1800** six-digit numbers in which only the digits 1, 3, 5, 7, and 9 appear, with each digit appearing at least once.

Would you like further clarification or details? 

Here are five related questions to consider:
1. How would the result change if the repeated digit could be chosen from a different set of digits?
2. What if the number needed to be divisible by 5? 
3. How would you calculate this if there were 7-digit numbers instead?
4. Can this problem be solved using combinations instead of permutations?
5. What happens if we allow the digits to repeat more than twice?

**Tip:** Always remember to account for repeating elements when using permutations.

Total number of 6-digit numbers in which only and all five digits 1, 3, 5, 7, and 9 appear is

Discover how to calculate the total number of 6-digit numbers that can be formed using the digits 1, 3, 5, 7, and 9, where each digit appears at least once. Learn the key combinatorial concepts and multiset permutation formulas to solve this problem step-by-step. Suitable for students in Grades 10-12.

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