To calculate the probability of winning the jackpot, we need to determine the total number of possible outcomes and then find the probability of selecting the correct outcome.

Step 1: Calculate the number of ways to choose 3 different numbers from 1 to 40

Since the order does not matter, we use the combination formula $C(n, k)$:

$C(40, 3) = \frac{40!}{3!(40 - 3)!} = \frac{40!}{3! \times 37!}$

Calculating this:

$C(40, 3) = \frac{40 \times 39 \times 38}{3 \times 2 \times 1} = 9,880$

Step 2: Calculate the number of ways to choose 1 number from 1 to 26

Since the order does not matter, and we are only choosing one number, the number of ways is simply 26.

Step 3: Calculate the total number of possible outcomes

The total number of possible outcomes is the product of the two calculated numbers:

$\text{Total outcomes} = C(40, 3) \times 26 = 9,880 \times 26 = 256,880$

Step 4: Calculate the probability of winning the jackpot

Since only one specific combination out of the total possible outcomes wins the jackpot, the probability of winning is:

$\text{Probability} = \frac{1}{256,880}$

Thus, the probability of winning the jackpot on one try is $\frac{1}{256,880}$.

Would you like to dive into more details or have any questions?

Here are 5 related questions:

1. How does the probability change if you select 4 numbers instead of 3 from the 40?
2. What is the probability of winning if the numbers were chosen in a specific order?
3. How does the jackpot probability compare to rolling a specific outcome on a 6-sided die?
4. What if the number range changes from 1 to 50 instead of 1 to 40—what’s the new probability?
5. What are the odds of winning the jackpot twice in a row?

Tip: When calculating probabilities in combinatorics, always check whether order matters. This affects whether you use combinations or permutations.