In a lottery game, a player picks 7 numbers from 1 to 41. How many different choices does the player have if order doesn't matter?

There are 
 choices to pick for the lottery

In this lottery game, the player picks 7 numbers from a set of 41, and the order in which the numbers are chosen doesn't matter. This is a **combination problem**, not a permutation problem, because the order is irrelevant.

The number of combinations of choosing \( r \) objects from a set of \( n \) objects is given by the formula:

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

Where:
- \( n = 41 \) (the total numbers to choose from),
- \( r = 7 \) (the number of numbers picked).

Substitute these values into the formula:

\[
\binom{41}{7} = \frac{41!}{7!(41 - 7)!} = \frac{41!}{7! \cdot 34!}
\]

Now, we calculate this value:

\[
\binom{41}{7} = \frac{41 \times 40 \times 39 \times 38 \times 37 \times 36 \times 35}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}
\]

Let's compute this.

The number of different choices the player has is \( 22,481,940 \).

Would you like more details or have any further questions? Here are five related questions that could expand your understanding:

1. What if the player picks 6 numbers instead of 7 from 1 to 41?
2. How many ways can a player pick 7 numbers from 1 to 50?
3. What is the probability of picking exactly one specific number in the lottery?
4. If the player picks 7 numbers from 1 to 41, how many different ways are there to arrange these 7 numbers in order?
5. How does the formula for combinations differ from that of permutations?

**Tip:** Combinations count selections where order does not matter, while permutations count selections where order does matter.

In a lottery game, a player picks 7 numbers from 1 to 41. How many different choices does the player have if order doesn't matter?

Discover how to calculate the number of combinations in a lottery where a player picks 7 numbers from 1 to 41. This problem uses the combination formula C(n, r) to find the total number of choices. Detailed explanation provided for high school students (Grades 10-12).

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