3.4 Combinations and Pascal's Triangle
Chapter
Chapter 3
Section
3.4
Solutions 27 Videos

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

0.30mins
Q1a

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

1.27mins
Q1b

Use Pascal’s method to complete the array. 0.48mins
Q2

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

0.41mins
Q3

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? 0.48mins
Q4

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

1.50mins
Q5

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.

1.18mins
Q6

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.

0.52mins
Q7a

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

0.41mins
Q7b

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.

1.09mins
Q8

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? 0.52mins
Q9

Determine the number of paths from A to B, travelling downward and to the right. 0.52mins
Q10

Determine the number of paths that spell PASCAL. Can combinations be used to solve each question? 0.59mins
Q11a

Determine the number of paths that spell PASCAL. Can combinations be used to solve each question? 0.37mins
Q11b

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.

2.02mins
Q12

Use Pascal’s triangle to expand and simplify.

\displaystyle (x + y)^8 

0.55mins
Q13a

Use Pascal’s triangle to expand and simplify.

\displaystyle (x - y)^5 

1.28mins
Q13b

Use Pascal’s triangle to expand and simplify.

\displaystyle (2a + b^4 

1.13mins
Q13c

Use Pascal’s triangle to expand and simplify.

\displaystyle (x^2 - 2)^3 

0.44mins
Q13d

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. 4.42mins
Q14a

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? 0.16mins
Q14b 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.

2.41mins
Q15

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?

2.40mins
Q16

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

2.16mins
Q17

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

1.54mins
Q18

Expand and simplify.

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

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