1. Copyright © 2003, N. Ahbel
Expansion of (x+y)n

2. (x + y) 0 = 1

3. (x + y) 0 = 1
(x + y) 1 = 1x1 + 1y1

4. (x + y) 0 = 1
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2

5. (x + y) 0 = 1
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3

6. (x + y) 0 = 1
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3
(x + y) 4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4

7. (x + y) 5 = 1x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1y5
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3
(x + y) 4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4
(x + y) 5 = 1x5 + 5x4y1 + 10x3y x2y3 + 5x1y4 + 1y5

8. (x + y) 5 = 1x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1y5
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3
(x + y) 4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4
(x + y) 5 = 1x5 + 5x4y1 + 10x3y x2y3 + 5x1y4 + 1y5
(x + y) 6 = 1x6 + 6x5y1 + 15x4y x3y x2y4 + 6x1y5 + 1y6

9. (x + y) 5 = 1x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1y5
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3
(x + y) 4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4
(x + y) 5 = 1x5 + 5x4y1 + 10x3y x2y3 + 5x1y4 + 1y5
(x + y) 6 = 1x6 + 6x5y1 + 15x4y x3y x2y4 + 6x1y5 + 1y6
(x + y) 7 = 1x7 + 7x6y1 + 21x5y x4y x3y x2y5 + 7x1y6 + 1y7

10. (x + y) 5 = 1x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1y5
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3
(x + y) 4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4
(x + y) 5 = 1x5 + 5x4y1 + 10x3y x2y3 + 5x1y4 + 1y5
(x + y) 6 = 1x6 + 6x5y1 + 15x4y x3y x2y4 + 6x1y5 + 1y6
(x + y) 7 = 1x7 + 7x6y1 + 21x5y x4y x3y x2y5 + 7x1y6 + 1y7
(x + y) 8 = 1x 8 + 8x7y1 + 28x6y x5y x4y x3y x2y6 + 8x1y7 + 1y8

11. (x + y) 5 = 1x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1y5
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3
(x + y) 4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4
(x + y) 5 = 1x5 + 5x4y1 + 10x3y x2y3 + 5x1y4 + 1y5
(x + y) 6 = 1x6 + 6x5y1 + 15x4y x3y x2y4 + 6x1y5 + 1y6
(x + y) 7 = 1x7 + 7x6y1 + 21x5y x4y x3y x2y5 + 7x1y6 + 1y7
(x + y) 8 = 1x 8 + 8x7y1 + 28x6y x5y x4y x3y x2y6 + 8x1y7 + 1y8
(x + y) 9 = 1x9 + 9x8y1 + 36x7y x6y x5y x4y x3y x2y7 + 9x1y8 + 1y9

12. (x + y) 5 = 1x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1y5
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3
(x + y) 4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4
(x + y) 5 = 1x5 + 5x4y1 + 10x3y x2y3 + 5x1y4 + 1y5
(x + y) 6 = 1x6 + 6x5y1 + 15x4y x3y x2y4 + 6x1y5 + 1y6
(x + y) 7 = 1x7 + 7x6y1 + 21x5y x4y x3y x2y5 + 7x1y6 + 1y7
(x + y) 8 = 1x 8 + 8x7y1 + 28x6y x5y x4y x3y x2y6 + 8x1y7 + 1y8
(x + y) 9 = 1x9 + 9x8y1 + 36x7y x6y x5y x4y x3y x2y7 + 9x1y8 + 1y9
(x + y) 10 = 1x x9y1 + 45x8y x7y x6y x5y x4y x3y7 + 45x2y8 + 10x1y9 + 1y10

13. (x + y) 5 = 1x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1y5
(x + y) 1 = 1x1 + 1y1
(x + y) 2 = 1x2 + 2x1y1 + 1y2
(x + y) 3 = 1x3 + 3x2y1 + 3x1y2 + 1y3
(x + y) 4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4
(x + y) 5 = 1x5 + 5x4y1 + 10x3y x2y3 + 5x1y4 + 1y5
(x + y) 6 = 1x6 + 6x5y1 + 15x4y x3y x2y4 + 6x1y5 + 1y6
(x + y) 7 = 1x7 + 7x6y1 + 21x5y x4y x3y x2y5 + 7x1y6 + 1y7
(x + y) 8 = 1x 8 + 8x7y1 + 28x6y x5y x4y x3y x2y6 + 8x1y7 + 1y8
(x + y) 9 = 1x9 + 9x8y1 + 36x7y x6y x5y x4y x3y x2y7 + 9x1y8 + 1y9
(x + y) 10 = 1x x9y1 + 45x8y x7y x6y x5y x4y x3y7 + 45x2y8 + 10x1y9 + 1y10

14. 1

15. 1
5+10=15

16. 1
21+7=28

17. There are many more patterns. What others can you find?1
There are many more patterns. What others can you find?

18. 1
1+4+10=15

19. 1
=36

20. 1
=23

21. 1
=24

22. The Fibonacci Sequence!1 2 3 5 8 13 21 34 55 89 1
The Fibonacci Sequence!

23. Arithmeticae in Numeris, 1631 William Oughtred (1574-1660)
7th row: 1,7,21,35,35,21,7,1

24. Pascal's
Triangle

25. Copyright © 2003, N. Ahbel
Expansion of (x+y)n