determine the odds of winning the lottery where there are 54 numbers, choose 5 and a bonus ball chosen from 15 numbers.

(54 over 5)

To determine the odds of winning this lottery, where you select 5 numbers from 54 and a bonus ball from 15, we need to calculate the number of possible combinations of the 5 numbers and multiply it by the number of possibilities for the bonus ball.

First, we calculate the number of combinations of choosing 5 numbers from 54. This is given by the binomial coefficient:

\[
\binom{54}{5} = \frac{54!}{5!(54-5)!} = \frac{54!}{5!49!}
\]

Now, let's compute the number of combinations:

\[
\binom{54}{5} = \frac{54 \times 53 \times 52 \times 51 \times 50}{5 \times 4 \times 3 \times 2 \times 1}
\]

Next, we multiply this by the number of ways to choose the bonus ball from 15 possibilities (which is simply 15).

The total number of possible outcomes is then:

\[
\binom{54}{5} \times 15
\]

I'll calculate this for you now.The total number of possible outcomes is **47,437,650**. This means the odds of winning the lottery, given you choose 5 numbers from 54 and 1 bonus ball from 15, are **1 in 47,437,650**.

Would you like more details or have any questions about the calculation?

Here are 5 related questions:
1. How does the inclusion of a bonus ball affect lottery odds?
2. What are the odds of winning if there were no bonus ball?
3. How do the odds change if we increase the number of numbers to choose from?
4. How can you calculate the probability of matching just the bonus ball?
5. What is the general formula for calculating lottery odds with multiple bonus balls?

**Tip:** Understanding the factorial function is key in combinatorics, as it helps calculate possible outcomes in probability.

Determine the odds of winning the lottery where there are 54 numbers, choose 5, and a bonus ball chosen from 15 numbers.

Learn how to calculate the odds of winning a lottery where 5 numbers are chosen from 54 and a bonus ball from 15. This combinatorics and probability-based problem involves calculating the binomial coefficient and understanding basic probability rules. Suitable for Grades 9-12.

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