**What would the odds of winning be if there are 54 numbers instead of 53 with one winning ticket?**

To determine the odds of winning a Pick 5 lottery game where players choose 5 out of 54 numbers, and there is only one winning ticket, we need to follow similar steps to the previous calculation.

### Step 1: Calculate the total number of possible combinations

The number of ways to choose 5 numbers out of 54 is given by the binomial coefficient:

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

### Step 2: Calculate the value of the binomial coefficient

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

Calculate the numerator first:

\[
54 \times 53 = 2862
\]

\[
2862 \times 52 = 148824
\]

\[
148824 \times 51 = 7580024
\]

\[
7580024 \times 50 = 379001200
\]

Now, calculate the denominator:

\[
5 \times 4 \times 3 \times 2 \times 1 = 120
\]

So,

\[
\binom{54}{5} = \frac{379001200}{120} = 3158343.33
\]

However, since we are dealing with combinations, it must be an integer:

\[
\binom{54}{5} = 3158343
\]

### Step 3: Determine the odds of winning with one winning ticket

If there is only one winning ticket, the odds of winning is the ratio of one winning combination to the total number of combinations:

\[
\text{Odds of winning} = \frac{1}{\binom{54}{5}} = \frac{1}{3158343}
\]

### Final Answer

The odds of winning the lottery, given there is one winning ticket, are:

\[
\text{Odds of winning} = \frac{1}{3158343}
\]

Would you like more details or have any questions?

Here are 5 questions you might ask next:
1. How does changing the number of total numbers in a lottery game affect the odds of winning?
2. What is the difference between permutations and combinations?
3. How can binomial coefficients be applied to other probability problems?
4. How do odds change with different lottery game formats (e.g., Pick 3, Pick 4)?
5. What are the strategies to calculate large binomial coefficients efficiently?

**Tip:** Understanding the difference between permutations and combinations is fundamental in combinatorics, especially in problems like lottery odds where order does not matter.

Learn how to calculate the odds of winning a Pick 5 lottery game with 54 numbers using combinatorial mathematics. Understand the binomial coefficient formula and its application in determining lottery odds. Suitable for high school students and lottery enthusiasts.

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