Number 12 3. Use the binomial theorem to find the coefficient of x³ in (x+2)134. JUse the binomial theorem to find the coefficient of x6 y in (3x- 2y).useful from time to time. For instance, you can use it if you ever(2a)* + 4(2a)°(-b)+6(2a)²(-bÝ + 4(2a)(-bỷ +(-b¢For now we will be content to accept the binomial theorem wis(You will be asked to prove it in an exercise in Chapter 10.) You mexpand an such as (x+ y)?. To do this, look at Row 1 t2. Use the binomial theorem to find the coefficient of x³ y³ in (x+ y)1310. Show that the formula k() = n(j) is true for all integers n,k with 0sksn.The Ineluston-43.7 The InelMany eountinof two finite sFirst wemust equalA and thenget IA1+Bcounted ea+7xyFor another example,(2a)+(-b))*(2a - b)4%3D16a –32a®b+24a²b² – 8ab³ + b4%3DThereforLAI+BFaIfExercises for Section 3.61. Write out Row 11 of Pascal's triangle.Noso Fato th5. Use the binomial theorem to show Eo4) = 2".6. Use Definition 3.2 (page 85) and Fact 1.3 (page 13) to show7. Use the binomial theorem to show E-, 3* () = 4".8. Use Fact 3.5 (page 87) to derive Equation 3.3 (page 90).9. Use the binomial theorem to show (6)–(1)+6)-(3)+(4) – …- -(-1)* (") =0, &formexclhin faCorExma11. Use the binomial theorem to show 9" = E"-,(-1)* (*)10"-k.12, Show that ()() = OC).13. Show that () = 6) + C) + () + () +14. The first five rows of Pascal's triangle appear in the digits of powers of 11:11-11 = 11, 112 = 121, 113 = 1331 and 114 = 14641, Why is this so? Why doestiepattern not continue with 11?th%3D(i• (";").+...+

Question

Number 12 3. Use the binomial theorem to find the coefficient of x³ in (x+2)134. JUse the binomial theorem to find the coefficient of x6 y in (3x- 2y).useful from time to time. For instance, you can use it if you ever(2a)* + 4(2a)°(-b)+6(2a)²(-bÝ + 4(2a)(-bỷ +(-b¢For now we will be content to accept the binomial theorem wis(You will be asked to prove it in an exercise in Chapter 10.) You mexpand an such as (x+ y)?. To do this, look at Row 1 t2. Use the binomial theorem to find the coefficient of x³ y³ in (x+ y)1310. Show that the formula k() = n(j) is true for all integers n,k with 0sksn.The Ineluston-43.7 The InelMany eountinof two finite sFirst wemust equalA and thenget IA1+Bcounted ea+7xyFor another example,(2a)+(-b))*(2a - b)4%3D16a –32a®b+24a²b² – 8ab³ + b4%3DThereforLAI+BFaIfExercises for Section 3.61. Write out Row 11 of Pascal's triangle.Noso Fato th5. Use the binomial theorem to show Eo4) = 2&#34;.6. Use Definition 3.2 (page 85) and Fact 1.3 (page 13) to show7. Use the binomial theorem to show E-, 3* () = 4&#34;.8. Use Fact 3.5 (page 87) to derive Equation 3.3 (page 90).9. Use the binomial theorem to show (6)–(1)+6)-(3)+(4) – …- -(-1)* (&#34;) =0, &formexclhin faCorExma11. Use the binomial theorem to show 9&#34; = E&#34;-,(-1)* (*)10&#34;-k.12, Show that ()() = OC).13. Show that () = 6) + C) + () + () +14. The first five rows of Pascal's triangle appear in the digits of powers of 11:11-11 = 11, 112 = 121, 113 = 1331 and 114 = 14641, Why is this so? Why doestiepattern not continue with 11?th%3D(i• (&#34;;&#34;).+...+

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Answered: Image /qna-images/answer/41c9a99d-b7a0-4fd5-bf33-176f05c46d40.jpg

12, Show that ()() = (OC)- n-m' %3D 13. Show thet (A)