Unit 6

Permutations, Combinations, and The Binomial Theorem

Introduction


Lesson 6.2: The Binomial Theorem

In previous units and lessons, you learned mathematicians, philosophers, artists, and scientists contributed to the fields of math you are learning today.

Blaise Pascal, a French mathematician, developed a triangular array of numbers that is related to the pathway problems you saw in Lesson 6.1.

In Lesson 6.2, you will


  • relate pathways and Pascal’s triangle to choosing «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math» elements from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» objects,
  • learn the relationship between Pascal’s triangle and the expansion of a binomial raised to the exponent «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math»,
  • expand binomials of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»n«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»N«/mi»«/mrow»«/mstyle»«/math» in various ways, including the binomial theorem, and
  • determine a specific term in the expansion of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»n«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»N«/mi»«/mrow»«/mstyle»«/math».

Coming into this Lesson, you should have


  • well-developed algebra skills,
  • familiarity with expanding a product of two or more polynomial factors, and
  • the ability to solve combinations problems.