Winning the jackpot in a particular lottery requires that you select the correct five numbers between 1 and 42 ​and, in a separate​ drawing, you must also select the correct single number between 1 and 43. Find the probability of winning the jackpot.
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Part 1
The probability of winning the jackpot is 
  
enter your response here.
​(Type an integer or simplified​ fraction.)


To find the probability of winning the jackpot in this lottery, we need to compute the probability of two independent events happening together:

1. **Selecting the correct five numbers** out of 42 numbers.
2. **Selecting the correct single number** out of 43 numbers.

### Step 1: Probability of selecting the correct five numbers
The order in which the five numbers are selected doesn't matter, so we use combinations. The number of ways to select 5 numbers from 42 is given by:

\[
\binom{42}{5} = \frac{42!}{5!(42-5)!} = \frac{42!}{5! \times 37!}
\]

This gives the total number of ways to choose 5 numbers out of 42.

### Step 2: Probability of selecting the correct single number
There are 43 possible numbers, and you need to pick the correct one. The probability of selecting this number is:

\[
\frac{1}{43}
\]

### Step 3: Calculating the overall probability
The overall probability is the product of the probability of selecting the correct 5 numbers and the probability of selecting the correct single number. This is:

\[
P = \frac{1}{\binom{42}{5}} \times \frac{1}{43}
\]

First, we calculate \( \binom{42}{5} \):

\[
\binom{42}{5} = \frac{42 \times 41 \times 40 \times 39 \times 38}{5 \times 4 \times 3 \times 2 \times 1} = 850668
\]

Now, the overall probability of winning the jackpot is:

\[
P = \frac{1}{850668} \times \frac{1}{43} = \frac{1}{36578724}
\]

Thus, the probability of winning the jackpot is:

\[
\boxed{\frac{1}{36578724}}
\]

Would you like more details or have any questions?

Here are five related questions:
1. What is the probability of selecting only the correct five numbers out of 42?
2. How does the probability change if the separate drawing has 50 numbers instead of 43?
3. What is the probability of selecting four correct numbers out of five in this lottery?
4. How many total combinations of five numbers can be chosen from 42?
5. What is the expected number of plays needed to win this lottery?

**Tip**: Remember, in probability, when dealing with independent events, their combined probability is the product of their individual probabilities.

Winning the jackpot in a particular lottery requires that you select the correct five numbers between 1 and 42 and, in a separate drawing, you must also select the correct single number between 1 and 43. Find the probability of winning the jackpot.

Find out how to calculate the probability of winning a lottery jackpot where you must choose 5 numbers from a pool of 42 and a separate single number from a pool of 43. This detailed explanation covers probability concepts, combinations, and step-by-step calculations. Suitable for high school students in Grades 11-12.

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