Math 1102 Activity 16: Combinations

Activity 16: Working with Combinations in Math 1102

Overview

• Focus on getting familiar with combinations through solving related problems.
• Collaboration is encouraged in team settings to enhance understanding.
• Steps should be documented clearly and shared digitally.

Pascal's Triangle

• Definition: Pascal's Triangle is a triangular array of the binomial coefficients. It starts with a single 1 at the top and each number is the sum of the two directly above it.
• Expansion: Typically starts with:
  • Row 0: 1
  • Row 1: 1 1
  • Row 2: 1 2 1
  • Row 3: 1 3 3 1
  • Row 4: 1 4 6 4 1
  • Row 5: 1 5 10 10 5 1

Task 1: Expanding Pascal's Triangle

1. Additional Row: Add Row 6:
   • Row 6: 1 6 15 20 15 6 1
2. Identifying Specific Combinations:
   • The value representing C(3, 2) in Pascal's Triangle is 3 (found in Row 3, Column 2).

Working with Combinations

Task 2: Evaluating Combinatorial Expressions

1. Calculating Values from Pascal's Triangle:
   • (a) C(5, 2) = 10
   • (b) C(5, 1) + C(5, 2) = 5 + 10 = 15
   • (c) 3! · C(5, 4) = 6 * 5 = 30 (since 3! = 6 and C(5, 4) = 5)
   • (d) C(4, 2) · C(4, 3) = 6 · 4 = 24 (where C(4, 2) = 6 and C(4, 3) = 4)

Problem-Solving Example

Task 3: Hiring Problem

• Situation: Hiring 2 people from a group of 6 applicants.
• Formula Used: The number of ways to choose k members from a group of n is given by the combination formula C(n, k) = n! / (k!(n-k)!).
• Calculation:
  • Here, n = 6 and k = 2:
  • C(6, 2) = 6! / (2!(6-2)!) = 6! / (2! * 4!)
  • Simplifying:
  • 6! = 720
  • 2! = 2
  • 4! = 24
  • Therefore:
  • C(6, 2) = 720 / (2 * 24) = 720 / 48 = 15
• Conclusion: There are 15 ways to hire 2 people from 6 applicants.

• Ensure all steps are clearly worked out and shared for group assignments as discussed in class. Use diagrams or visuals where beneficial to illustrate concepts.