# Worksheet: Pascal's Triangle and the Binomial Theorem

In this worksheet, we will practice using Pascal’s triangle to find the coefficients of the algebraic expansion of any binomial expression of the form (a+b)ⁿ.

Q1:

Daniel has been exploring the relationship between Pascalโs triangle and the binomial expansion. He has noticed that each row of Pascalโs triangle can be used to determine the coefficients of the binomial expansion of , as shown in the figure. For example, the fifth row of Pascalโs triangle can be used to determine the coefficients of the expansion of .

By calculating the next row of Pascalโs triangle, find the coefficients of the expansion of .

• A1, 7, 21, 34, 34, 21, 7, 1
• B1, 7, 21, 35, 35, 21, 7, 1
• C1, 6, 8, 20, 8, 1
• D2, 8, 12, 8, 2
• E1, 6, 15, 20, 15, 6, 1

Daniel now wants to calculate the coefficients for each of the terms of the expansion . By substituting into the expression above, or otherwise, calculate all of the coefficients of the expansion.

• A16, 32, 24, 8, 1
• B8, 32, 24, 8, 1
• C16, 64, 24, 8, 1
• D8, 64, 24, 8, 1
• E8, 32, 24, 8, 1

Q2:

Find the coefficient of in the expansion of .

Q3:

Matthew knows that he can use the row of Pascalโs triangle to calculate the coefficients of the expansion .

Calculate the numbers in the row of Pascalโs triangle and, hence, write out the coefficients of the expansion .

• A1, 6, 15, 20, 15, 6, 1
• B1, 5, 10, 10, 5, 1
• C1, 7, 21, 35, 35, 21, 7, 1
• D2, 6, 15, 20, 15, 6, 2
• E1, 3, 3, 1

Now, by considering the different powers of and and using Pascalโs triangle, work out the coefficients of the expansion .

• A32, , 320, , 160,
• B , 160, , 320, , 32
• C32, 160, 320, 320, 160, 32
• D64, 160, 320, 320, 160, 64
• E64, , 320, , 160,

Q4:

Shown is a partially filled-in picture of Pascalโs triangle. By spotting patterns, or otherwise, find the values of , , , and .

• A , , , and
• B , , , and
• C , , , and
• D , , , and
• E , , , and

Q5:

Fully expand the expression .

• A
• B
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• D
• E

Q6:

Find the coefficient of in the expansion of .

Q7:

Use Pascalโs triangle to expand the expression .

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• E

Q8:

Use Pascalโs triangle to expand the expression .

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• E

Q9:

Use Pascalโs triangle to expand the expression .

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• E

Q10:

Write the first 5 terms of the expansion of in ascending powers of .

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• E

Q11:

In the expansion of a binomial, determine which of the following is equivalent to the relation .

• A
• B
• C
• D

Q12:

Use Pascalโs triangle to determine the coefficients of the terms that result from the expansion of .

• A
• B
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• E