Unit 6

Permutations, Combinations, and The Binomial Theorem

Combinations and Binomial Expansions

Another interesting connection to Pascal’s triangle is the values in each row correspond to the values determined by a pattern of combinations.


By putting the pattern of the values in the triangle together with the combinations sequence, a method for determining the values in any given row of the triangle was developed.

Prior to this discovery, the method for finding the values in a line of Pascal’s triangle was recursive. For example, the values in row «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» could not be determined without knowing the values in row «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math», etc.

However, with the connection to combinations, it is no longer necessary to complete the triangle to know the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»5«/mn»«mi»th«/mi»«/msup»«/mstyle»«/math» row contains the values that start with «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«/mstyle»«/math», and there will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» values in that row.

Likewise, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»10«/mn»«mi»th«/mi»«/msup»«/mstyle»«/math» row will start with «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»9«/mn»«none/»«/mmultiscripts»«/mstyle»«/math» and there will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» values in the row.

Putting the combinations in place of the values in the triangle means any value of the triangle can be determined by considering in which row of the triangle the value lies.

The combinations then correspond to the values of the coefficients in a binomial expansion.

For example, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mn»4«/mn»«/msup»«/mstyle»«/math» corresponds to the fifth row of the triangle, or if you are referring to the triangle with combinations, it is the row that starts with «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«/mstyle»«/math».

The coefficients of the terms in the expansion will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»1«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»2«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»and«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»4«/mn»«none/»«mprescripts/»«mn»4«/mn»«none/»«/mmultiscripts»«/mrow»«/mstyle»«/math», which we can read from the table above. However, if we are required to expand a binomial raised to an exponent greater than the number of rows shown in the tables above, we can determine the coefficients from the combinations.

For example, the binomial «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mn»7«/mn»«/msup»«/mstyle»«/math» could be expanded by performing multiplication of


Recall «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mi»y«/mi»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».

Therefore, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mspace linebreak=¨newline¨/»«mtable columnalign=¨left¨»«mtr»«mtd»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mi»y«/mi»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mi»y«/mi»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mi»y«/mi»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/mrow»«/mstyle»«/math»

Further multiplication over the brackets follows one step at a time. However, this method is long and may lead to errors because of the many steps involved. Consider the triangle made from combinations.

The coefficients of the expansion correspond to the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»8«/mn»«mi»th«/mi»«/msup»«/mstyle»«/math» row of Pascal’s triangle, or the row starting with «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«/mstyle»«/math». The row will contain values that correspond to the combinations «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mmultiscripts»«mi»C«/mi»«mn»0«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»1«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»2«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»4«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»5«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»6«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»and«/mi»«mspace width=¨0.33em¨/»«mmultiscripts»«mi»C«/mi»«mn»7«/mn»«none/»«mprescripts/»«mn»7«/mn»«none/»«/mmultiscripts»«/mrow»«/mstyle»«/math».

Also, each term contains variables whose exponents add to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»7«/mn»«/mstyle»«/math».


Now, replace the combinations with the numerical coefficients.