exam_i

Exam I Review

Topics

• Combinatorics including
  • Dice
  • Card Hands
• • Binomial Expansion
  • Geometric Series
  • Exponential Series
  • Logrithmic Series
• • Binomial coefficients
  • Multinomial coefficients
• • Axioms of probability
  • Conditional probability
• Baye's formula
• Independence
• Inclusion-exclusion
• • Expectation
  • Variance
• • Binomial random variable [(n,p)]
  • Poisson random variable [(\lambda)]
  • Geometric random variable [(p)]
• • Stirling's Formula
  • Integrals of probability theory

Formulas

Chapter 1: Combinatorial Analysis

Binomial Coefficient

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

where [0 \le i \le n]

Binomial Theorem

[(x + y)^n = \sum\limits_{i = 0}^{n}{ {n \choose i} x^i y^{n-i}}]

Multinomial Coefficient

For nonnegative integers [n_1, \dots, n_r] summing to [n]

[{n \choose n_1, n_2, \dots, n_r} = \frac{ n!}{ n_1! n_2! \cdots n_r!}]

is the number of division of [n] items into [r] distinct nonoverlapping subgroups of sizes [n_1, n_2, \dots, n_r]

Problems

Chapter 1: Combinitorial Analysis

Problem 3

[20] workers are to be assigned to [20] different jobs, one to each job. How many different assignments are possible?

• [20] unique jobs and [20] unique workers [\Rightarrow 20!]

Problem 7

In how many ways can 3 boys and 3 girls sit in a row?

• [6] unique individuals [\Rightarrow 6!]

In how many ways can [3] boys and [3] girls sit in a row if the boys and the girls are each to sit together?

• [2] groups of [3] unique individuals [\Rightarrow 3! 3!]

In how many ways if only the boys must sit together?

• [3! {6 \choose 3}]