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, 35, 35, 21, 7, 1
  • B2, 8, 12, 8, 2
  • C1, 7, 21, 34, 34, 21, 7, 1
  • D1, 6, 15, 20, 15, 6, 1
  • E1, 6, 8, 20, 8, 1

Daniel now wants to calculate the coefficients for each of the terms of the expansion (2๐‘ฅ+๐‘ฆ)๏Šช. By substituting 2๐‘ฅ into the expression above, or otherwise, calculate all of the coefficients of the expansion.

  • A8, 64, 24, 8, 1
  • B8, 32, 24, 8, 1
  • C16, 64, 24, 8, 1
  • D16, 32, 24, 8, 1
  • E8, 32, 24, 8, 1

Q2:

Find the coefficient of ๐‘Ž๏Šซ in the expansion of ๏€ผ๐‘Ž+1๐‘Ž๏ˆ๏€ผ๐‘Ž+1๐‘Ž๏ˆ๏Šจ๏Šจ๏Šฉ๏Šฉ.

Q3:

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

Calculate the numbers in the 6th 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
  • B2, 6, 15, 20, 15, 6, 2
  • C1, 7, 21, 35, 35, 21, 7, 1
  • D1, 5, 10, 10, 5, 1
  • 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 (2๐‘Žโˆ’2๐‘)๏Šซ.

  • A32, โˆ’160, 320, โˆ’320, 160, โˆ’32
  • B64, 160, 320, 320, 160, 64
  • C32, 160, 320, 320, 160, 32
  • D64, โˆ’160, 320, โˆ’640, 160, โˆ’64
  • Eโˆ’32, 160, โˆ’320, 320, โˆ’160, 32

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‘‘.

  • A๐‘Ž=6, ๐‘=11, ๐‘=21, and ๐‘‘=10
  • B๐‘Ž=6, ๐‘=11, ๐‘=15, and ๐‘‘=10
  • C๐‘Ž=9, ๐‘=12, ๐‘=21, and ๐‘‘=10
  • D๐‘Ž=10, ๐‘=15, ๐‘=21, and ๐‘‘=11
  • E๐‘Ž=10, ๐‘=15, ๐‘=20, and ๐‘‘=11

Q5:

Fully expand the expression (2+3๐‘ฅ)๏Šง๏Šฆ.

  • A1,024+7,680๐‘ฅ+34,560๐‘ฅ+103,680๐‘ฅ+217,728๐‘ฅ+326,592๐‘ฅ+349,920๐‘ฅ+524,880๐‘ฅ+393,660๐‘ฅ+196,830๐‘ฅ+59,049๐‘ฅ๏Šจ๏Šฉ๏Šช๏Šซ๏Šฌ๏Šญ๏Šฎ๏Šฏ๏Šง๏Šฆ
  • B1,024+15,360๐‘ฅ+34,560๐‘ฅ+46,080๐‘ฅ+40,320๐‘ฅ+24,192๐‘ฅ+10,080๐‘ฅ+2,880๐‘ฅ+540๐‘ฅ+60๐‘ฅ+3๐‘ฅ๏Šจ๏Šฉ๏Šช๏Šซ๏Šฌ๏Šญ๏Šฎ๏Šฏ๏Šง๏Šฆ
  • C1,024+15,360๐‘ฅ+103,680๐‘ฅ+414,720๐‘ฅ+1,088,640๐‘ฅ+1,959,552๐‘ฅ+2,449,440๐‘ฅ+2,099,520๐‘ฅ+1,180,980๐‘ฅ+393,660๐‘ฅ+59,049๐‘ฅ๏Šจ๏Šฉ๏Šช๏Šซ๏Šฌ๏Šญ๏Šฎ๏Šฏ๏Šง๏Šฆ
  • D1,024+5,120๐‘ฅ+11,520๐‘ฅ+15,360๐‘ฅ+13,440๐‘ฅ+8,064๐‘ฅ+3,360๐‘ฅ+960๐‘ฅ+180๐‘ฅ+20๐‘ฅ+๐‘ฅ๏Šจ๏Šฉ๏Šช๏Šซ๏Šฌ๏Šญ๏Šฎ๏Šฏ๏Šง๏Šฆ
  • E10,240+69,120๐‘ฅ+276,480๐‘ฅ+725,760๐‘ฅ+1,306,368๐‘ฅ+1,632,960๐‘ฅ+1,399,680๐‘ฅ+787,320๐‘ฅ+262,440๐‘ฅ+39,366๐‘ฅ+59,049๐‘ฅ๏Šจ๏Šฉ๏Šช๏Šซ๏Šฌ๏Šญ๏Šฎ๏Šฏ๏Šง๏Šฆ

Q6:

Find the coefficient of ๐‘ฅ๏Šซ in the expansion of (2๐‘ฅ+5)๏Šง๏Šจ.

Q7:

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

  • A๐‘ฅ+3๐‘ฅ๐‘ฆ+6๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • B๐‘ฅ+3๐‘ฅ๐‘ฆ+9๐‘ฅ๐‘ฆ+3๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • C๐‘ฅ+4๐‘ฅ๐‘ฆ+9๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช
  • D๐‘ฅ+4๐‘ฅ๐‘ฆ+6๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šจ๏Šจ๏Šจ๏Šฉ๏Šช
  • E๐‘ฅ+4๐‘ฅ๐‘ฆ+6๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ+๐‘ฆ๏Šช๏Šฉ๏Šจ๏Šจ๏Šฉ๏Šช

Q8:

Use Pascalโ€™s triangle to expand the expression ๏€ผ๐‘ฅ+1๐‘ฅ๏ˆ๏Šช.

  • A๐‘ฅ+4๐‘ฅ+6+4๐‘ฅ+1๐‘ฅ๏Šช๏Šจ๏Šจ๏Šช
  • B๐‘ฅ+4๐‘ฅ+6+4๐‘ฅ+1๐‘ฅ๏Šช๏Šฉ๏Šช
  • C๐‘ฅ+4๐‘ฅ+6+4๐‘ฅ+1๐‘ฅ๏Šช๏Šฉ๏Šจ๏Šช
  • D๐‘ฅ+4๐‘ฅ+6+1๐‘ฅ+1๐‘ฅ๏Šช๏Šจ๏Šจ๏Šช
  • E๐‘ฅ+6๐‘ฅ+6+4๐‘ฅ+1๐‘ฅ๏Šช๏Šจ๏Šจ๏Šช

Q9:

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

  • A๐‘ฅ+12๐‘ฅ+54๐‘ฅ+108๐‘ฅ๏Šช๏Šฉ๏Šจ
  • B๐‘ฅ+12๐‘ฅ+54๐‘ฅ+90๐‘ฅ+81๏Šช๏Šฉ๏Šจ
  • C๐‘ฅ+12๐‘ฅ+54๐‘ฅ+108๐‘ฅ+81๏Šช๏Šฉ๏Šจ
  • D๐‘ฅ+9๐‘ฅ+81๐‘ฅ+81๐‘ฅ+81๏Šช๏Šฉ๏Šจ
  • E๐‘ฅ+4๐‘ฅ+18๐‘ฅ+36๐‘ฅ+27๏Šช๏Šฉ๏Šจ

Q10:

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

  • A262,144+2,359,296๐‘ฅ+10,027,008๐‘ฅ+26,738,688๐‘ฅ+50,135,040๐‘ฅ๏Šจ๏Šฉ๏Šช
  • B4,718,592+20,054,016๐‘ฅ+53,477,376๐‘ฅ+100,270,080๐‘ฅ+280,756,224๐‘ฅ๏Šจ๏Šฉ๏Šช
  • C262,144+4,718,592๐‘ฅ+40,108,032๐‘ฅ+213,909,504๐‘ฅ+802,160,640๐‘ฅ๏Šจ๏Šฉ๏Šช
  • D262,144+2,228,224๐‘ฅ+8,912,896๐‘ฅ+22,282,240๐‘ฅ+38,993,920๐‘ฅ๏Šจ๏Šฉ๏Šช
  • E262,144+1,179,648๐‘ฅ+3,342,336๐‘ฅ+6,684,672๐‘ฅ+10,027,008๐‘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‘Žcoe๏ฌƒcientofcoe๏ฌƒcientofcoe๏ฌƒcientof๏Šง๏Šฆ๏Šฏ๏Šง๏Šง.

  • A2(๐‘Ž)=๐‘Ž+๐‘Žcoe๏ฌƒcientofcoe๏ฌƒcientofcoe๏ฌƒcientof๏Šช๏Šซ๏Šฉ
  • B2(๐‘Ž)=๐‘Ž+๐‘Žcoe๏ฌƒcientofcoe๏ฌƒcientofcoe๏ฌƒcientof๏Šญ๏Šฎ๏Šฌ
  • C2(๐‘Ž)=๐‘Ž+๐‘Žcoe๏ฌƒcientofcoe๏ฌƒcientofcoe๏ฌƒcientof๏Šฌ๏Šญ๏Šซ
  • D2(๐‘Ž)=๐‘Ž+๐‘Žcoe๏ฌƒcientofcoe๏ฌƒcientofcoe๏ฌƒcientof๏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.

  • A1,6,15,20,15,6,1
  • B1,6,7,13,7,6,1
  • C1,3,6,10,15,21,28
  • D1,6,6,6,6,6,1
  • E1,5,10,10,5,1

Q13:

Find the coefficient of ๐‘Ž๏Šซ in the expansion of (9๐‘ฅ+2)๏Šฌ.

Q14:

Find the coefficient of ๐‘ฅ๏Šญ in the expansion of ๏€ผ2+3๐‘ฅ5๏ˆ๏Šง๏Šง.

  • A46,189,440
  • B8,44815,625
  • C2,18778,125
  • D9,237,88815,625
  • E2,309,47215,625

Q15:

Determine the coefficient of ๐‘Ž๏Šจ in the expansion of ๏€ผ๐‘Ž11+116๐‘Ž๏ˆ๏Šง๏Šจ.

  • A59,504
  • B11,188
  • C77,128
  • D0

Q16:

Write the coefficients of the terms that result from the expansion of (๐‘ฅ+๐‘ฆ)๏Šฉ.

  • A1,3,3,1
  • B3,6,3
  • C1,2,2,1
  • D3,6,6,3
  • E1,4,6,4,1

Q17:

Find the product of the coefficients of the terms of the expansion of (1โˆ’๐‘ฅ)๏Šฉ.

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