12, Show that ()() = (OC)- n-m' %3D 13. Show thet (A)
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Number 12
Transcribed Image Text:3. Use the binomial theorem to find the coefficient of x³ in (x+2)13
4. 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 m
expand an such as (x+ y)?. To do this, look at Row 1 t
2. Use the binomial theorem to find the coefficient of x³ y³ in (x+ y)13
10. Show that the formula k() = n(j) is true for all integers n,k with 0sks
n.
The Ineluston-4
3.7 The Inel
Many eountin
of two finite s
First we
must equal
A and then
get IA1+B
counted ea
+7xy
For another example,
(2a)+(-b))*
(2a - b)4
%3D
16a –32a®b+24a²b² – 8ab³ + b4
%3D
Therefor
LAI+B
Fa
If
Exercises for Section 3.6
1. Write out Row 11 of Pascal's triangle.
No
so Fa
to th
5. Use the binomial theorem to show Eo4) = 2".
6. Use Definition 3.2 (page 85) and Fact 1.3 (page 13) to show
7. 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, &
form
exclh
in fa
Cor
Ex
ma
11. 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 doestie
pattern not continue with 11?
th
%3D
(i
• (";").
+...+
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