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Lesson Worksheet: Pascal’s Triangle and the Binomial Theorem Mathematics

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:

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

Q2:

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

Q3:

Michael 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

Michael 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

Q4:

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

Q5:

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.

Q6:

Find the coefficient of ๐‘ฅ๏Šซ in the expansion of (2โˆ’5๐‘ฅ)๏Šฎ.

Q7:

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

Q8:

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

Q9:

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

Q10:

In the expansion of (2๐‘ฅ+5๐‘ฆ)๏Šซ according to the ascending power of ๐‘ฆ, if the two middle terms are equal, find the value of ๐‘ฅ๐‘ฆ.

  • A254
  • B52
  • C25
  • D32
  • E425

This lesson includes 23 additional questions and 155 additional question variations for subscribers.

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