How many combinations are possible with 10 initial choices and 5 decisions to make if
repetition is allowed?

If repetition is allowed, we can use the formula for permutations with repetition. When making \( n \) decisions from \( k \) choices, the number of combinations is given by:

\[ k^n \]

Here, we have:
- \( k = 10 \) initial choices
- \( n = 5 \) decisions to make

Using the formula:

\[ 10^5 = 100,000 \]

So, there are 100,000 possible combinations when making 5 decisions from 10 choices with repetition allowed.

Would you like more details on this topic, or do you have any other questions?

Here are 8 relative questions for further exploration:
1. How do you calculate combinations without repetition?
2. What is the difference between permutations and combinations?
3. How would the number of combinations change if we had 8 initial choices instead of 10?
4. How do you calculate permutations with a subset of the total choices?
5. How do combinations with repetition differ from combinations without repetition?
6. What are some real-world applications of permutations and combinations?
7. How do you calculate the number of permutations with restrictions (e.g., certain elements must be in specific positions)?
8. Can you explain the binomial coefficient and its role in combinations?

**Tip:** When dealing with large numbers of combinations or permutations, using factorials can simplify calculations.

Understand how to calculate combinations with repetition using 10 initial choices and making 5 decisions. Learn the formula for permutations with repetition and its application in combinatorics. Suitable for students and enthusiasts interested in mathematical permutations and combinations.

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