"Counting Principle"

Committee Selection Overview

• **Positions to be filled:

  • Faculty Member
  • Male Student
  • Female Student
  • Parent**

• Participants Interested in Each Position:

  • Faculty Members: 2 (Mrs. Baker, Ms. Perry)
  • Male Students: 4 (Kevin, David, Hans, John)
  • Female Students: 4 (Ashley, Yoko, Kaitlin, Mai)
  • Parents: 3 (Ms. Chen, Mr. Adams, Mrs. Perez)

Fundamental Counting Principle

• Definition:

  • The fundamental counting principle states that if there are $N<em>1$ ways to do one thing and $N</em>2$ ways to do another, then there are $N<em>1 imes N</em>2$ ways to do both.

• Application:

  • In this context, to find the total number of ways to fill the committee positions, multiply the number of options for each position:

  $ext{Total Ways} = N<em>{faculty} imes N</em>{male} imes N<em>{female} imes N</em>{parent}$

  Calculating the Total Options

• Substitute the numbers into the formula:

  • $N_{faculty} = 2$

  • $N_{male} = 4$

  • $N_{female} = 4$

  • $N_{parent} = 3$

  • Thus, the calculation becomes:

  $ext{Total Ways} = 2 imes 4 imes 4 imes 3$

• Calculation:

  • Compute step by step:
  • First: $2 imes 4 = 8$
  • Then: $8 imes 4 = 32$
  • Finally: $32 imes 3 = 96$

• Final Answer:

  • There are 96 different ways to fill the four committee positions.

Importance of Combinations

• This example illustrates the practical applications of combinations and probabilistic calculations in real-life scenarios such as committee formation.

• Understanding the fundamental counting principle is essential for solving various problems in combinatorics, probability, and statistics.