1. Blaise Pascal “We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others. Pensées “(1670) “Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.”

2. 11 1 Pascal's Triangle 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

3. 11 1 Zeroth Row  1st Row  1 2 1 2nd Row  1 3 3 1 3rd Row  1 4 6 4 1 4th Row  1 5 10 10 5 1 1 6 15 20 15 6 1

4. 4C4 4C0 4C1 4C2 4C3 11 1 Binomial Coeff. for a group of 4 1 2 1 1 3 3 1 1 4 6 4 1 4th  1 5 10 10 5 1

5. Binomial Coeff. for a group of 4 11 1 1 2 1 1 3 3 1 These are the coefficients in the expansion of (x+y)4 1 4 6 4 1 4th  1 5 10 10 5 1

6. Other Interesting Patterns 11 1 1 2 1 Counting Numbers 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

7. Other Interesting Patterns 11 1 1 2 1 Triangular Numbers 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

8. Other Interesting Patterns 11 1 sum = 1 Powers of 2 sum = 2 1 2 1 sum = 4 1 3 3 1 sum = 8 1 4 6 4 1 sum = 16 1 5 10 10 5 1 sum = 32 1 6 15 20 15 6 1 sum = 64

9. 11 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Other Interesting Patterns sum = 2 sum = 3 sum = 5 sum = 8 sum = 13 Fibonacci Numbers

10. Other Interesting Patterns Pascal’s Flowers The gray cell is surrounded by 6 “petals.” If you multiply the yellow petals you get 2100. If you multiply the orange petals, you get 2100.

11. Other Interesting Patterns 11 1  11 0  11 1 1 2 1  11 2 Powers of 11 1 3 3 1  11 3 1 4 6 4 1  11 4 1 5 10 10 5 1 1 6 15 20 15 6 1