Suppose the Lottery Commission offers a Pick
5
game, where the players choose
5
out of
40
numbers. What would the odds of winning be if there are three winning tickets in this drawing?

To determine the odds of winning a Pick 5 lottery game where players choose 5 out of 40 numbers, and there are three winning tickets, we need to follow these steps:

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

The number of ways to choose 5 numbers out of 40 is given by the binomial coefficient:

\[
\binom{40}{5} = \frac{40!}{5!(40-5)!}
\]

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

\[
\binom{40}{5} = \frac{40!}{5! \cdot 35!}
\]

First, calculate the factorials involved:

- \(40! = 40 \times 39 \times 38 \times \cdots \times 1\)
- \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
- \(35! = 35 \times 34 \times 33 \times \cdots \times 1\)

We only need the product of the first 5 numbers from 40 factorial since the rest will cancel out with \(35!\):

\[
40 \times 39 \times 38 \times 37 \times 36
\]

\[
= 40 \times 39 = 1560
\]

\[
1560 \times 38 = 59280
\]

\[
59280 \times 37 = 2193360
\]

\[
2193360 \times 36 = 78960960
\]

Now, divide by \(5!\):

\[
\binom{40}{5} = \frac{78960960}{120} = 658008
\]

So, the total number of combinations is 658,008.

### Step 3: Determine the odds of winning with three winning tickets

If there are three winning tickets, then out of all possible combinations, three are winners. The odds of winning is the ratio of winning combinations to total combinations:

\[
\text{Odds of winning} = \frac{3}{\binom{40}{5}} = \frac{3}{658008}
\]

This fraction can be simplified as:

\[
\text{Odds of winning} = \frac{3}{658008} \approx 4.56 \times 10^{-6}
\]

### Final Answer

The odds of winning the lottery, given there are three winning tickets, are approximately:

\[
\text{Odds of winning} \approx \frac{1}{219336}
\]

So, the odds of winning are 1 in 219,336.

Would you like more details or have any questions?

Here are 5 questions you might ask next:
1. How are binomial coefficients used in probability calculations?
2. What is the factorial function, and how is it applied in combinatorics?
3. How do you simplify large factorial calculations in combinatorics problems?
4. What are the odds of winning if there is only one winning ticket in the Pick 5 lottery?
5. Can you explain how the odds change with a different number of winning tickets?

**Tip:** In probability, understanding how to calculate combinations is crucial for solving lottery and other similar problems.

Understand how to calculate the odds of winning in a Pick 5 lottery game where players choose 5 out of 40 numbers, with three winning tickets. Learn about combinatorics and probability calculations involved in determining the likelihood of winning. Suitable for advanced high school students and anyone interested in probability theory.

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