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 ( ๐‘ฅ + ๐‘ฆ ) ๏Š , 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 ( ๐‘ฅ + ๐‘ฆ ) ๏Šช .

By calculating the next row of Pascalโ€™s triangle, find the coefficients of the expansion of ( ๐‘ฅ + ๐‘ฆ ) ๏Šฌ .

  • 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 ( 2 ๐‘ฅ + ๐‘ฆ ) ๏Šช . By substituting 2 ๐‘ฅ 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 ๐‘Ž ๏Šซ in the expansion of ๏€ผ ๐‘Ž + 1 ๐‘Ž ๏ˆ ๏€ผ ๐‘Ž + 1 ๐‘Ž ๏ˆ ๏Šจ ๏Šจ ๏Šฉ ๏Šฉ .

Q3:

Matthew knows that he can use the 6 t h row of Pascalโ€™s triangle to calculate the coefficients of the expansion ( ๐‘Ž + ๐‘ ) ๏Šซ .

Calculate the numbers in the 6 t h row of Pascalโ€™s triangle and, hence, write out the coefficients of the expansion ( ๐‘Ž + ๐‘ ) ๏Šซ .

  • 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 ๐‘Ž and ๐‘ and using Pascalโ€™s triangle, work out the coefficients of the expansion ( 2 ๐‘Ž โˆ’ 2 ๐‘ ) ๏Šซ .

  • A32, โˆ’ 1 6 0 , 320, โˆ’ 3 2 0 , 160, โˆ’ 3 2
  • B โˆ’ 3 2 , 160, โˆ’ 3 2 0 , 320, โˆ’ 1 6 0 , 32
  • C32, 160, 320, 320, 160, 32
  • D64, 160, 320, 320, 160, 64
  • E64, โˆ’ 1 6 0 , 320, โˆ’ 6 4 0 , 160, โˆ’ 6 4

Q4:

Shown is a partially filled-in picture of Pascalโ€™s triangle. By spotting patterns, or otherwise, find the values of ๐‘Ž , ๐‘ , ๐‘ , and ๐‘‘ .

  • A ๐‘Ž = 1 0 , ๐‘ = 1 5 , ๐‘ = 2 1 , and ๐‘‘ = 1 1
  • B ๐‘Ž = 9 , ๐‘ = 1 2 , ๐‘ = 2 1 , and ๐‘‘ = 1 0
  • C ๐‘Ž = 6 , ๐‘ = 1 1 , ๐‘ = 1 5 , and ๐‘‘ = 1 0
  • D ๐‘Ž = 6 , ๐‘ = 1 1 , ๐‘ = 2 1 , and ๐‘‘ = 1 0
  • E ๐‘Ž = 1 0 , ๐‘ = 1 5 , ๐‘ = 2 0 , and ๐‘‘ = 1 1

Q5:

Fully expand the expression ( 2 + 3 ๐‘ฅ ) ๏Šง ๏Šฆ .

  • A 1 , 0 2 4 + 7 , 6 8 0 ๐‘ฅ + 3 4 , 5 6 0 ๐‘ฅ + 1 0 3 , 6 8 0 ๐‘ฅ + 2 1 7 , 7 2 8 ๐‘ฅ + 3 2 6 , 5 9 2 ๐‘ฅ + 3 4 9 , 9 2 0 ๐‘ฅ + 5 2 4 , 8 8 0 ๐‘ฅ + 3 9 3 , 6 6 0 ๐‘ฅ + 1 9 6 , 8 3 0 ๐‘ฅ + 5 9 , 0 4 9 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช ๏Šซ ๏Šฌ ๏Šญ ๏Šฎ ๏Šฏ ๏Šง ๏Šฆ
  • B 1 , 0 2 4 + 5 , 1 2 0 ๐‘ฅ + 1 1 , 5 2 0 ๐‘ฅ + 1 5 , 3 6 0 ๐‘ฅ + 1 3 , 4 4 0 ๐‘ฅ + 8 , 0 6 4 ๐‘ฅ + 3 , 3 6 0 ๐‘ฅ + 9 6 0 ๐‘ฅ + 1 8 0 ๐‘ฅ + 2 0 ๐‘ฅ + ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช ๏Šซ ๏Šฌ ๏Šญ ๏Šฎ ๏Šฏ ๏Šง ๏Šฆ
  • C 1 , 0 2 4 + 1 5 , 3 6 0 ๐‘ฅ + 3 4 , 5 6 0 ๐‘ฅ + 4 6 , 0 8 0 ๐‘ฅ + 4 0 , 3 2 0 ๐‘ฅ + 2 4 , 1 9 2 ๐‘ฅ + 1 0 , 0 8 0 ๐‘ฅ + 2 , 8 8 0 ๐‘ฅ + 5 4 0 ๐‘ฅ + 6 0 ๐‘ฅ + 3 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช ๏Šซ ๏Šฌ ๏Šญ ๏Šฎ ๏Šฏ ๏Šง ๏Šฆ
  • D 1 0 , 2 4 0 + 6 9 , 1 2 0 ๐‘ฅ + 2 7 6 , 4 8 0 ๐‘ฅ + 7 2 5 , 7 6 0 ๐‘ฅ + 1 , 3 0 6 , 3 6 8 ๐‘ฅ + 1 , 6 3 2 , 9 6 0 ๐‘ฅ + 1 , 3 9 9 , 6 8 0 ๐‘ฅ + 7 8 7 , 3 2 0 ๐‘ฅ + 2 6 2 , 4 4 0 ๐‘ฅ + 3 9 , 3 6 6 ๐‘ฅ + 5 9 , 0 4 9 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช ๏Šซ ๏Šฌ ๏Šญ ๏Šฎ ๏Šฏ ๏Šง ๏Šฆ
  • E 1 , 0 2 4 + 1 5 , 3 6 0 ๐‘ฅ + 1 0 3 , 6 8 0 ๐‘ฅ + 4 1 4 , 7 2 0 ๐‘ฅ + 1 , 0 8 8 , 6 4 0 ๐‘ฅ + 1 , 9 5 9 , 5 5 2 ๐‘ฅ + 2 , 4 4 9 , 4 4 0 ๐‘ฅ + 2 , 0 9 9 , 5 2 0 ๐‘ฅ + 1 , 1 8 0 , 9 8 0 ๐‘ฅ + 3 9 3 , 6 6 0 ๐‘ฅ + 5 9 , 0 4 9 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช ๏Šซ ๏Šฌ ๏Šญ ๏Šฎ ๏Šฏ ๏Šง ๏Šฆ

Q6:

Find the coefficient of ๐‘ฅ ๏Šซ in the expansion of ( 2 ๐‘ฅ + 5 ) ๏Šง ๏Šจ .

Q7:

Use Pascalโ€™s triangle to expand the expression ( ๐‘ฅ + ๐‘ฆ ) ๏Šช .

  • A ๐‘ฅ + 4 ๐‘ฅ ๐‘ฆ + 6 ๐‘ฅ ๐‘ฆ + 4 ๐‘ฅ ๐‘ฆ + ๐‘ฆ ๏Šช ๏Šฉ ๏Šจ ๏Šจ ๏Šฉ ๏Šช
  • B ๐‘ฅ + 4 ๐‘ฅ ๐‘ฆ + 6 ๐‘ฅ ๐‘ฆ + 4 ๐‘ฅ ๐‘ฆ + ๐‘ฆ ๏Šช ๏Šจ ๏Šจ ๏Šจ ๏Šฉ ๏Šช
  • C ๐‘ฅ + 3 ๐‘ฅ ๐‘ฆ + 9 ๐‘ฅ ๐‘ฆ + 3 ๐‘ฅ ๐‘ฆ + ๐‘ฆ ๏Šช ๏Šฉ ๏Šจ ๏Šจ ๏Šฉ ๏Šช
  • D ๐‘ฅ + 3 ๐‘ฅ ๐‘ฆ + 6 ๐‘ฅ ๐‘ฆ + 4 ๐‘ฅ ๐‘ฆ + ๐‘ฆ ๏Šช ๏Šฉ ๏Šจ ๏Šจ ๏Šฉ ๏Šช
  • E ๐‘ฅ + 4 ๐‘ฅ ๐‘ฆ + 9 ๐‘ฅ ๐‘ฆ + 4 ๐‘ฅ ๐‘ฆ + ๐‘ฆ ๏Šช ๏Šฉ ๏Šจ ๏Šจ ๏Šฉ ๏Šช

Q8:

Use Pascalโ€™s triangle to expand the expression ๏€ผ ๐‘ฅ + 1 ๐‘ฅ ๏ˆ ๏Šช .

  • A ๐‘ฅ + 4 ๐‘ฅ + 6 + 4 ๐‘ฅ + 1 ๐‘ฅ ๏Šช ๏Šฉ ๏Šจ ๏Šช
  • B ๐‘ฅ + 4 ๐‘ฅ + 6 + 1 ๐‘ฅ + 1 ๐‘ฅ ๏Šช ๏Šจ ๏Šจ ๏Šช
  • C ๐‘ฅ + 6 ๐‘ฅ + 6 + 4 ๐‘ฅ + 1 ๐‘ฅ ๏Šช ๏Šจ ๏Šจ ๏Šช
  • D ๐‘ฅ + 4 ๐‘ฅ + 6 + 4 ๐‘ฅ + 1 ๐‘ฅ ๏Šช ๏Šฉ ๏Šช
  • E ๐‘ฅ + 4 ๐‘ฅ + 6 + 4 ๐‘ฅ + 1 ๐‘ฅ ๏Šช ๏Šจ ๏Šจ ๏Šช

Q9:

Use Pascalโ€™s triangle to expand the expression ( 3 + ๐‘ฅ ) ๏Šช .

  • A ๐‘ฅ + 1 2 ๐‘ฅ + 5 4 ๐‘ฅ + 1 0 8 ๐‘ฅ ๏Šช ๏Šฉ ๏Šจ
  • B ๐‘ฅ + 1 2 ๐‘ฅ + 5 4 ๐‘ฅ + 1 0 8 ๐‘ฅ + 8 1 ๏Šช ๏Šฉ ๏Šจ
  • C ๐‘ฅ + 4 ๐‘ฅ + 1 8 ๐‘ฅ + 3 6 ๐‘ฅ + 2 7 ๏Šช ๏Šฉ ๏Šจ
  • D ๐‘ฅ + 1 2 ๐‘ฅ + 5 4 ๐‘ฅ + 9 0 ๐‘ฅ + 8 1 ๏Šช ๏Šฉ ๏Šจ
  • E ๐‘ฅ + 9 ๐‘ฅ + 8 1 ๐‘ฅ + 8 1 ๐‘ฅ + 8 1 ๏Šช ๏Šฉ ๏Šจ

Q10:

Write the first 5 terms of the expansion of ( 2 + ๐‘ฅ ) ๏Šง ๏Šฎ in ascending powers of ๐‘ฅ .

  • A 2 6 2 , 1 4 4 + 1 , 1 7 9 , 6 4 8 ๐‘ฅ + 3 , 3 4 2 , 3 3 6 ๐‘ฅ + 6 , 6 8 4 , 6 7 2 ๐‘ฅ + 1 0 , 0 2 7 , 0 0 8 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช
  • B 4 , 7 1 8 , 5 9 2 + 2 0 , 0 5 4 , 0 1 6 ๐‘ฅ + 5 3 , 4 7 7 , 3 7 6 ๐‘ฅ + 1 0 0 , 2 7 0 , 0 8 0 ๐‘ฅ + 2 8 0 , 7 5 6 , 2 2 4 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช
  • C 2 6 2 , 1 4 4 + 2 , 3 5 9 , 2 9 6 ๐‘ฅ + 1 0 , 0 2 7 , 0 0 8 ๐‘ฅ + 2 6 , 7 3 8 , 6 8 8 ๐‘ฅ + 5 0 , 1 3 5 , 0 4 0 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช
  • D 2 6 2 , 1 4 4 + 4 , 7 1 8 , 5 9 2 ๐‘ฅ + 4 0 , 1 0 8 , 0 3 2 ๐‘ฅ + 2 1 3 , 9 0 9 , 5 0 4 ๐‘ฅ + 8 0 2 , 1 6 0 , 6 4 0 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช
  • E 2 6 2 , 1 4 4 + 2 , 2 2 8 , 2 2 4 ๐‘ฅ + 8 , 9 1 2 , 8 9 6 ๐‘ฅ + 2 2 , 2 8 2 , 2 4 0 ๐‘ฅ + 3 8 , 9 9 3 , 9 2 0 ๐‘ฅ ๏Šจ ๏Šฉ ๏Šช

Q11:

In the expansion of a binomial, determine which of the following is equivalent to the relation 2 ( ๐‘Ž ) = ๐‘Ž + ๐‘Ž c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f ๏Šง ๏Šฆ ๏Šฏ ๏Šง ๏Šง .

  • A 2 ( ๐‘Ž ) = ๐‘Ž + ๐‘Ž c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f ๏Šช ๏Šซ ๏Šฉ
  • B 2 ( ๐‘Ž ) = ๐‘Ž + ๐‘Ž c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f ๏Šญ ๏Šฎ ๏Šฌ
  • C 2 ( ๐‘Ž ) = ๐‘Ž + ๐‘Ž c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f ๏Šฌ ๏Šญ ๏Šซ
  • D 2 ( ๐‘Ž ) = ๐‘Ž + ๐‘Ž c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f ๏Šซ ๏Šฌ ๏Šช

Q12:

Use Pascalโ€™s triangle to determine the coefficients of the terms that result from the expansion of ( ๐‘ฅ + ๐‘ฆ ) ๏Šฌ .

  • A 1 , 3 , 6 , 1 0 , 1 5 , 2 1 , 2 8
  • B 1 , 5 , 1 0 , 1 0 , 5 , 1
  • C 1 , 6 , 6 , 6 , 6 , 6 , 1
  • D 1 , 6 , 7 , 1 3 , 7 , 6 , 1
  • E 1 , 6 , 1 5 , 2 0 , 1 5 , 6 , 1

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