In a certain lottery, 5 different numbers between 1 and 36 inclusive are drawn, and these are the winning numbers. To win the lottery, a person must select the correct 5 numbers in the same order in which they are drawn. What is the probability of winning if one ticket is purchased?

To determine the probability of winning the lottery, we need to calculate how many possible ways the 5 numbers can be selected in the correct order, and then the probability of choosing the correct combination.

### Step 1: Total number of possible outcomes
In this lottery, 5 different numbers are drawn in a specific order from a set of 36 numbers. This means we are dealing with a **permutation** problem, since the order matters.

The number of possible ways to select and arrange 5 numbers from 36 is given by the formula for permutations:

\[
P(36, 5) = \frac{36!}{(36-5)!} = \frac{36!}{31!}
\]

This simplifies to:

\[
P(36, 5) = 36 \times 35 \times 34 \times 33 \times 32
\]

### Step 2: Calculate the total number of permutations
Now we compute the total number of permutations:

\[
36 \times 35 \times 34 \times 33 \times 32 = 45,239,040
\]

### Step 3: Probability of winning
The probability of selecting the correct 5 numbers in the correct order on a single ticket is:

\[
\text{Probability} = \frac{1}{45,239,040}
\]

Thus, the probability of winning the lottery by purchasing one ticket is:

\[
\boxed{\frac{1}{45,239,040}}
\]

This is an extremely small probability, showing how unlikely it is to win.

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

Here are some related questions for further practice:
1. What if the numbers could be repeated in the lottery draw? How would that change the probability?
2. What is the probability of winning if you buy 10 tickets?
3. How does the probability change if only 4 numbers are drawn instead of 5?
4. What if the order of the numbers doesn’t matter, how does this affect the probability?
5. What is the probability of matching exactly 4 out of 5 numbers in the correct order?

**Tip:** When calculating probabilities, always consider if the order of events matters. If order matters, you're dealing with permutations, otherwise, it's a combination.

Calculate the probability of winning a lottery where 5 numbers are drawn between 1 and 36 in a specific order. Learn how to apply the permutation formula and probability rules to determine the odds of winning. Suitable for high school students in Grades 9-12.

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