Around the middle of the 17th century, Blaise Pascal described a certain triangle of numbers that had applications in the theory of probability. Though this triangle has been named in honor of Pascal, it was described several centuries earlier by the Chinese mathematician Yanghui and by the Arabian mathematician Omar Khayyám. In this project, you will study Pascal's Triangle.

1. Consider the experiment of flipping three fair coins.
   1. Find the sample space for this experiment. How many outcomes are in the sample space?
   2. In how many different ways can one obtain 0 heads? 1 head? 2 heads? 3 heads?
   3. What is the probability of obtaining 0 heads? 1 head? 2 heads? 3 heads?
2. Repeat the problem above for the experiment of flipping four fair coins.
3. We use the notation C(n,k) to denote the number of ways of choosing k items from n distinct items. It turns out that C(n,k) = n!/( k!(n-k)!).
   1. Compute C(4,3), C(3,2), and C(6,4).
   2. Describe the numbers computed in Problems 1b and 2b in terms of this new notation.
4. Write a one or two paragraph biography of Blaise Pascal.
5. On the attached sheet, form thirteen rows of a triangle of numbers such that the top row consists of a 1, the next row consists of two 1s, and subsequent rows are formed by adding numbers in the previous row. For instance, the top four rows of the triangle should look like this:

   1 1 1     1 2 1     1 3 3 1

6. The previous problem introduced you to Pascal's Triangle. Typically, the top row is called row 0 and the leftmost element in each row is called element 0. The rest of the numbers in the triangle are labeled accordingly. For instance, element 1 in row 2 is 2 and element 2 in row 3 is 3.
   1. Using this numbering scheme, describe the numbers in Pascal's Triangle using the notation developed in Problem 3. Specifically, find a formula for element p in row m.
   2. Find a formula for the sum of the numbers in row k.
7. Use Pascal's Triangle to calculate the following probabilities.
   1. The probability of obtaining 4 heads if 7 coins are flipped.
   2. The probability of obtaining 8 heads if 10 coins are flipped.
   3. The probability of obtaining 10 heads if 12 coins are flipped.
8. The entries in Pascal's Triangle are sometimes called binomial coefficients. Clearly explain why.
9. The Fibonacci sequence appears in Pascal's triangle. Where? Describe another pattern in Pascal's triangle.