Write each term in row 9 of Pascal’s triangle using _nC_r

Write the first five terms in diagonal 4 of Pascal’s triangle using _nC_r

Use Pascal’s method to complete the array.

Find the first three terms in the expansion of xI + y)^7.

Point B is four blocks east and three blocks south of point A. How many routes are possible from A to B, travelling only south and east?

Write the first nine rows of Pascal’s triangle. Circle the given terms in Pascal's triangle, then circle the correct answer to help use Pascal’s method to rewrite each of the following:

a) _5C_2+_5C_3

b) _7C_3+_7C_4

c) _5C_4- _4C_4

d) _8C_6- _7C_5

a) Evaluate each of the following:

• i) _2C_2 + _3C_2
• ii) _3C_2 + _4C_2
• iii) _4C_2 + _5C_2

b) Describe the results.

c) Identify the terms from part a) in Pascal's triangle.

Calculate the sum of the first four terms of diagonal 7 (diagonals begin at zero). Locate the sum in Pascal's triangle and relate it to _nC_r.

Generalize by stating the sum of the first k terms of diagonal r in Pascal's triangle and relating it to _nC_r

A checker is placed on a checkerboard in the top right corner. The checker can move diagonally downward. Determine the number of routes to the bottom of the board.

A black checker is placed in the bottom-right corner of a checkerboard. The checker can move diagonally upward. The black checker cannot enter the square occupied by the red checker, but can jump over it. How many routes are there for the black checker to the top of the board?

Determine the number of paths from A to B, travelling downward and to the right.

Determine the number of paths that spell PASCAL. Can combinations be used to solve each question?

Determine the number of paths that spell PASCAL. Can combinations be used to solve each question?

Investigate the sums of the first 72 natural numbers. For example, 1+2=3,1+2+3=6,...

a) Locate the results in Pascal’s triangle.

b) Summarize the results using combinations.

Use Pascal’s triangle to expand and simplify.

\displaystyle (x + y)^8

Use Pascal’s triangle to expand and simplify.

\displaystyle (x - y)^5

Use Pascal’s triangle to expand and simplify.

\displaystyle (2a + b^4

Use Pascal’s triangle to expand and simplify.

\displaystyle (x^2 - 2)^3

Drawing a line through a circle divides it into two regions.

• If 71 lines are drawn through a circle, what is the maximum number of regions formed? Develop a formula using Pascal’s triangle.

Drawing a line through a circle divides it into two regions.

• Twenty lines are drawn through a circle. What is the maximum number of regions inside the circle?

a) Investigate the sum of the squares of the natural numbers from 1 to n by copying and completing the chart up to n = 6.

b) Describe the results.

c) Generalize using combinations.

d) State the sum of the squares of the first 50 natural numbers.

Investigate the number of oranges needed to stack the fruit in a tetrahedron.

a) Complete a chart showing the total number of oranges needed relative to the number of layers.

b) Identify the results in Pascal’s triangle and describe your findings.

c) Write a relationship involving _nC_r

d) How many oranges are needed for a 10-layer stack in a tetrahedral shape?

a Calculate the sum of the squares of the terms in rows 2 to 5 of Pascal’s Triangle.

b) Identify the sums from part a) in Pascal’s triangle. Describe your findings.

c) Rewrite each of the sums using combinations.

d) Generalize the relationship using a formula involving _nC_r

Explain how (h + t)^5 could be used to show the different combinations of heads and tails when a coin is tossed repeatedly.

Expand and simplify.

\displaystyle (p - \frac{1}{p})^5

Expand and simplify.

\displaystyle (3m^2 | + \frac{2}{m^2})^4