Learning Resources

Lesson

Although the remainder of the work in this unit is not directly related to probability, it is related to combinations which we studied in order to calculate probability.

Pascal's Triangle is a triangular array of numbers that has many patterns imbedded within it. In this lesson we will look at only a few of these. The main pattern in which we are interested is the one we will use to construct the "triangle". This pattern is that "any element in the triangle is the sum of the two elements above it , immediately to the left and to the right". Try to determine the number in each cell in the 2nd row yourself by applying the pattern stated above. After you figure out what the number should be, click on each ? mark to see if your are correct. Then repeat the same for row 3, 4, etc.

Row 0 1 2 3 4 5 6 7

The Investigation in your text asks you to find "as many patterns as you can" in the triangle. One possibility you might consider is to investigate the diagonals. Use the above diagram, reset everything to ? marks and then highlight the first diagonal. Is there a pattern? Repeat the process and highlight the second diagonal. What pattern does this show? What about the other diagonals?

There are many other patterns. Share what you find with the other members of your class (either on site or via the net). See if you can find a pattern that your friends did not see.

It is a good idea to complete Pascal's Triangle to row 10 for future reference. To help you to do this, ask you on site teacher to get you Blackline Master 5.5.1 from the Teacher's Resource Binder. Fill it in and place it in your notebook.

At the beginning of the lesson we said that Pascal's Triangle could be related to combinations.

To see the connection between combinations and Pascal's Triangle we need to first reconsider factorial notation again. We have defined n! as follows:

n! = n x (n - 1) x (n - 2) x ... x 3 x 2 x 1

1! should be fairly obvious. It is 1. But what about 0!? How do we calculate it or does it even exist? By necessity, mathematicians have defined 0! as 1. Note that this is a definition and no proof of it is required or possible.

Suppose we consider the expression _nC _r and for n use the row number of Pascal's Triangle and for r the element number in that row (remember we start counting at zero for both the rows and the elements). Let's calculate a combination and compare it to the number in Pascal's triangle.

Use the triangle below and go to row 3. Click on element number 2 in this row (remember start counting at 0). What is the value? How does it compare to ₃C₂?

Row 0 1 2 3 4 5 6 7

Lets try another combination.

Use the triangle above and go to row 6. Click on element number 4 in this row (again remember to start counting at 0). What is the value? How does it compare to ₆C₄?

Make a conjecture. Evaluate _nC _r for several other values of n and r and use the triangle to check your conjecture.

The relationship between the elements of Pascal's Triangle and combinations is summarized below.

The elements in Pascal's Triangle follow the pattern shown below when they are written using combinations:

Row 0 1 2 3 4

This information gives a convenient pattern for finding any element of the triangle without completing the table up to the row containing the required element. This will be of advantage to us in the next lesson dealing with the expansion of binomials.

You should now be ready to complete the assigned work on this lesson.

Activity

1. Complete Investigation 12 on page 340 in your text.
2. Do the CYU questions 1 & 2 on pages 40 & 341.
3. Complete Investigation 13 on page 341.
4. Complete the Investigation Question 3 on page 341.
5. Do The CYU Questions 4 - 6, 8 on pages 341 & 342.

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

Use combinations to find the following elements in Pascal's Triangle:

1. The first three elements in row 9.
2. The 6th element in row 10.

Solution