1. Lesson 4

1.8. Explore 4

Mathematics 30-1 Module 8

Module 8: Permutations, Combinations, and the Binomial Theorem

 

The value of an entry in Pascal’s triangle can be determined using combinations. nCk is equal to the value of the (k + 1)th number of the (n + 1)th row. For example, in 5C4, k = 4 and n = 5. So, you need to look to the (4 + 1)th, or 5th, number in the (5 + 1)th, or 6th, row. So, 5C4 = 5.

 

The first diagram shows the first 6 rows of Pascal’s triangle with the fifth number of the sixth row highlighted. The second diagram shows the first six rows of Pascal’s triangle represented with combinations. The fifth combination in the sixth row is highlighted.

 

So far you have seen a pattern for determining

  • the exponents of a binomial expansion
  • a method for determining the coefficients of a binomial expansion using Pascal’s triangle
  • a method of determining the values of Pascal’s triangle using combinations

Putting these together results in the binomial theorem, which can be used to determine each term of a binomial expansion. You may have already determined your own version of the binomial theorem in Try This 3. One common way of writing the theorem is as follows:

 

 

(x + y)n = nC0(x)n(y)0 + nC1(x)n−1(y)1 + nC2(x)n−2(y)2 + + nCn−1(x)1(y)n−1 + nCn(x)0(y)n

 

Try to match the terms of this expression to the expression you began to expand in Try This 3.

 

 

 

If you are interested in a particular term, the following formula can be used:

 

 

tk+1 = nCk(x)n−k(y)k

 

Watch Term of a Binomial Expansion to see how this formula is used to determine the sixth term of (3f − g2)14.

 

 

This play button opens Term of a Binomial Expansion.


textbook

If you would like to see another example of how the binomial theorem can be used, read “Example 2” on page 540 of the textbook. Otherwise, continue to Self-Check 2.

 

 

Self-Check 2
  1. Complete “Your Turn” from “Example 2” on page 541 of the textbook. Answer
  2. Complete questions 3.b., 11.c., 12.a., 12.b., 17.a., and 17.d. on pages 542 to 543 of the textbook. Answer

glossary

Add the following terms to your copy of Glossary Terms:

  • Pascal’s triangle
  • binomial theorem

formula

Add the following formulas to your copy of Formula Sheet:

  • binomial theorem:

     
    (x + y)n = nC0(x)n(y)0 + nC1(x)n−1(y)1 + nC2(x)n−2(y)2 + + nCn−1(x)1(y)n−1 + nCn(x)0(y)n
  • general term in the expansion of (x + y)n: tk + 1 = nCk (x)n − k (y)k