3. (a) suppose you toss a coin 43 times, how many ways can you get 17 heads (in any order)?
(b) Suppose you toss a coin 43 times, how many ways can you get 17 heads, where there is one head tossed first, and one head tossed last?


### Part (a)

To determine how many ways you can get 17 heads in 43 coin tosses (in any order), we use the concept of combinations. The number of ways to choose 17 heads out of 43 tosses is given by the combination formula:

\[
\text{Number of ways} = \binom{43}{17} = \frac{43!}{17!(43-17)!}
\]

This represents the number of combinations of 43 items taken 17 at a time.

### Part (b)

For part (b), we want to find the number of ways to get 17 heads, with the additional condition that one head must be tossed first and one head must be tossed last.

1. First, consider that the first and last tosses are heads. This uses up 2 of the 17 heads, so we are left with 15 heads to place among the remaining 41 tosses.
2. The number of ways to choose 15 positions out of the remaining 41 for these heads is given by:

\[
\text{Number of ways} = \binom{41}{15} = \frac{41!}{15!(41-15)!}
\]

This represents the number of ways to distribute the remaining 15 heads in 41 tosses, given that the first and last tosses are already heads.

### Summary of Answers:

- **(a) Number of ways to get 17 heads in 43 tosses (any order):** \(\binom{43}{17}\)
- **(b) Number of ways to get 17 heads with heads on the first and last tosses:** \(\binom{41}{15}\)

Would you like the exact numerical values for these combinations or any further details? 

Here are five related questions:

1. How would the problem change if we wanted exactly 20 heads instead of 17?
2. What is the probability of getting exactly 17 heads in 43 tosses?
3. How would the solution change if the coin was biased, with a different probability for heads?
4. How would you calculate the expected number of heads in 43 coin tosses?
5. What if you wanted at least 17 heads? How would you calculate that?

**Tip:** When dealing with combinations in probability, remember that \(\binom{n}{k}\) represents the number of ways to choose \(k\) successes out of \(n\) trials, regardless of the order.

3. (a) Suppose you toss a coin 43 times, how many ways can you get 17 heads (in any order)? (b) Suppose you toss a coin 43 times, how many ways can you get 17 heads, where there is one head tossed first, and one head tossed last?

Explore how to calculate the number of ways to get 17 heads in 43 coin tosses using combinatorics. This detailed solution covers the combination formula for both general cases and specific conditions. Ideal for high school students in Grades 10-12.

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