Winning the jackpot in a particular lottery requires that you select the correct four numbers between 1 and 31 ​and, in a separate​ drawing, you must also select the correct single number between 1 and 23. Find the probability of winning the jackpot.

To find the probability of winning the jackpot in this lottery, we need to calculate the total number of possible outcomes and the number of favorable outcomes (which is just 1, since only one specific combination wins the jackpot).

### Steps:

1. **Selecting the 4 numbers between 1 and 31:**
   The order in which the 4 numbers are selected does not matter, so we need to calculate the number of combinations (not permutations) of 4 numbers chosen from a set of 31 numbers. This is given by the combination formula:

   \[
   \binom{31}{4} = \frac{31!}{4!(31-4)!} = \frac{31 \times 30 \times 29 \times 28}{4 \times 3 \times 2 \times 1} = 31 \times 30 \times 29 \times 28 \div 24 = 31,465
   \]

2. **Selecting the correct single number between 1 and 23:**
   There are 23 possible numbers, so the number of ways to select one correct number from 23 is simply 23.

3. **Total possible outcomes:**
   The total number of possible outcomes for the lottery is the product of the two independent events: choosing the 4 numbers from 31 and choosing the single number from 23:

   \[
   31,465 \times 23 = 723,695
   \]

4. **Favorable outcome:**
   Since there is only one specific combination of numbers that will win, the number of favorable outcomes is 1.

5. **Probability of winning the jackpot:**
   The probability of winning is the ratio of favorable outcomes to the total possible outcomes:

   \[
   P(\text{winning}) = \frac{1}{723,695}
   \]

Thus, the probability of winning the jackpot is:

\[
\boxed{\frac{1}{723,695}}
\]

### Do you want more details or have any questions?

Here are some related questions:
1. How do you calculate the probability for other lottery systems with different rules?
2. What if the order of numbers matters in the drawing? How would that change the probability?
3. How does the probability change if fewer numbers are drawn, such as 3 out of 31?
4. What is the expected value of a lottery ticket given the probability and the prize amount?
5. Can you explain why we use combinations instead of permutations in this lottery problem?

**Tip:** When dealing with probabilities, always check whether the order of selection matters (permutations) or not (combinations).

Winning the jackpot in a particular lottery requires that you select the correct four numbers between 1 and 31 and, in a separate drawing, you must also select the correct single number between 1 and 23. Find the probability of winning the jackpot.

Explore how to calculate the probability of winning a lottery where you choose 4 numbers from 31 and 1 number from 23. This problem introduces key concepts in combinatorics and probability, providing a step-by-step solution suitable for high school students in Grades 10-12.

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