1. Combinations and Pascal’s triangle
Miniconference on the Mathematics of Computation
MTH 210
Combinations and Pascal’s triangle
Dr. Anthony Bonato
Ryerson University

2. Combinations n non-negative integer, r ≤ n
𝑛 𝑟 = number of ways to choose r objects from n objects
say: “n choose r”
call 𝑛 𝑟 a combination or binomial coefficient
can show: 𝑛 𝑟 = 𝑛! 𝑛−𝑟 !𝑟!
NB: 0! = 1

3. Pascal’s triangle …

4. Pascal’s triangle …

5. Pascal’s triangle
nth row:
𝑛 𝑛 𝑛 2 … 𝑛 𝑛− 𝑛 𝑛
1 n n(n-1)/ n

6. Key facts
𝑛 𝑟 = 𝑛 𝑛−𝑟
𝑛+1 𝑟 = 𝑛 𝑟−1 + 𝑛 𝑟

7. Binomial theorem
x, y variables, n non-negative integer

8. Exercises

9. Miniconference on the Mathematics of Computation
MTH 210
Asymptotics I
Dr. Anthony Bonato
Ryerson University

10. Asymptotics we consider the growth rate of functions
y = f(x) when x gets large
need a way to compare growth rates of different functions y = f(x) and y = g(x)

11. Example
y=f(x) = x + 1, y = g(x) = x2 - 5x - 17

12. Big Oh notation
assume y = f(x) and y = g(x) are real-valued functions, and lim 𝑥→∞ 𝑓 𝑥 𝑔 𝑥 exists and is finite
we write f(x) = O(g(x))
also say f(x) is in O(g(x))
say: “f(x) is big Oh of g(x)”

13. Key facts for all r < s, xr = O(xs)
anxn+an-1xn-1 + … + a1x +a0 = O(xn), where an > 0
if f(x) = O(g(x)) and g(x) = O(h(x)), then f(x) = O(h(x)) (transitive property)

14. Exercises