In a certain​ lottery, an urn contains balls numbered 1 to 28. From this​ urn, 6 balls are chosen​ randomly, without replacement. For a​ $1 bet, a player chooses one set of six numbers. To​ win, all six numbers must match those chosen from the urn. The order in which the balls are selected does not matter. What is the probability of winning this lottery with one​ ticket?
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Part 1
The probability of winning is 
  
enter your response here.
​(Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to the nearest hundredth as​ needed.)

The problem describes a lottery where 6 balls are chosen randomly from an urn containing 28 balls, and the player must match all 6 numbers to win. To find the probability of winning, we will calculate the number of possible sets of 6 balls that can be chosen from the 28 balls and then determine the probability of picking the correct set.

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

Since the order in which the balls are selected does not matter, we use the combination formula:

\[
C(n, r) = \frac{n!}{r!(n - r)!}
\]

Where:
- $ n = 28 $ is the total number of balls.
- $ r = 6 $ is the number of balls chosen.

\[
C(28, 6) = \frac{28!}{6!(28 - 6)!} = \frac{28!}{6!22!}
\]

This gives the total number of different combinations of 6 balls that can be selected from the 28.

### Step 2: Calculate the probability

The probability of winning with one ticket is the ratio of the number of winning outcomes (which is 1, since only one specific set of 6 numbers will win) to the total number of possible outcomes:

\[
\text{Probability} = \frac{1}{C(28, 6)}
\]

Now let's calculate that:

\[
C(28, 6) = \frac{28 \times 27 \times 26 \times 25 \times 24 \times 23}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 376740
\]

Thus, the probability of winning is:

\[
\text{Probability} = \frac{1}{376740}
\]

### Step 3: Express in scientific notation

The probability is:

\[
\text{Probability} \approx 2.65 \times 10^{-6}
\]

So, the probability of winning this lottery with one ticket is approximately $ 2.65 \times 10^{-6} $.

Would you like further details or have any questions? Here are some related questions:

1. How would the probability change if there were 30 balls instead of 28?
2. What is the formula for permutations, and how does it differ from combinations?
3. If 3 balls out of 6 need to match, what is the probability of winning?
4. How does probability change when repetition is allowed in selection?
5. Can you calculate the probability of winning in a Powerball lottery?

**Tip:** Always use combinations instead of permutations when the order of selection doesn't matter.

In a certain lottery, an urn contains balls numbered 1 to 28. From this urn, 6 balls are chosen randomly, without replacement. For a $1 bet, a player chooses one set of six numbers. To win, all six numbers must match those chosen from the urn. The order in which the balls are selected does not matter. What is the probability of winning this lottery with one ticket?

Find the probability of winning a lottery where 6 balls are drawn randomly from an urn of 28 balls. This detailed solution explains how to calculate the total number of combinations and the probability, using combinatorics and probability theory. Suitable for students in Grades 10-12.

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