lecture_02

Mon Jun 11 14:00:34 CDT 2012

For Wed

pg.16 # 3, 7, 8, 14 pg.18 # 8, 9, 12, 13

Read 1.1 - 1.5

Binomial Theorem

[{\left( {1 + x} \right)^h} = \sum\limits_{k = 0}^n {n \choose k} x^k ]

[ {n \choose k} = \frac{ n!}{ k!( n - k)!}]

[{\left( {1 + x} \right)^h} = (1 + x)(1 + 3x + 3x^2 + x^3)]

[=] [1] [3x] [3x^2] [x^3]

Prop.

[ {n \choose k} + { n \choose k+1} = {n+1 \choose k+1}]

[ \frac{ n!}{ k!( n-k)!} + \frac{ n!}{ (k+1)!(n-k-1)!} = \frac{ n!}{k!(n-k-1)!} \left( \frac{ 1}{ n-k} + \frac{ 1}{ k+1} \right)]

[ \frac{ (n+1)!}{ (k+1)!(n-k)! } = {n+1 \choose k+1}]

History

Geometry

2600 years ago: many facts

"theorems from axioms"

2300 years ago (Euclid).

Probability

Accurate facts about odds in gambling known around 17th centry

"theorems from axioms" about 1930 (kolmogovov)