Explore

Pascal’s Triangle Solutions

Pascal’s Triangle is a triangular arrangement of numbers in which each number is the sum of the two numbers directly above it.
Mathematicians have found many applications for Pascal’s Triangle, including:
- Finding the number of combinations for a problem
- Determining the coefficients of n^th row binomial expressions
- Finding the sum of n-rows using 2ⁿ
Pascal’s Triangle is constructed by calculating the sum of the two numbers in the row directly above it. 
- Each row starts and ends with one . 
- The top row of the triangle is “ row zero .”
- In each row that follows, the second number represents which row it is (row 1, row 2, etc.).

Note

For example, row 2 (1 2 1) shows you there’s one way to choose zero items from a set of two, two ways to choose one item, and one way to choose both items.

To determine a combination, nCr, using Pascal’s Triangle:
- Go to the n^th row
- Count over to the (r + 1)^th term (the first term will always be the left-most “1” in the row)
- Record the answer

For this lesson, a Pascal’s Triangle handout has been provided for you. You should be able to create additional rows on your own using the established pattern.

Example 1

Use Pascal’s Triangle to evaluate.

$n C r (13, 7) = 1, 716$

Explain

The ${(r + 1)}^{t h}$ term in Pascal’s Triangle has the answer to the combination $(7 + 1 = 8^{t h}$ term)
Find the $8^{t h}$ term in the $13^{t h}$ row

$n C r (15, 11) = 1, 365$

Note

$11 + 1 = 12^{t h}$ term

Find the $12^{t h}$ term in the $15^{t h}$ row

Example 2

Use Pascal’s Triangle to determine the sum of the row.

Row 6

$2^{6} = 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64$

Row 17

$2^{17} = 131, 072$