${(a+b)^0=1}$

$(a+b)^1=1 \cdot a+1 \cdot b$

$(a+b)^2=1 \cdot a^2 +2ab + 1 \cdot b^2$

$(a+b)^3=1 \cdot a^3+3 a^2 b+3 a b^2+b^3$

$(a+b)^7=(a+b) \cdot(a+b) \cdot(a+b) \cdot(a+b) \cdot(a+b) \cdot(a+b) \cdot(a+b)$

$=a^7+7 a^6 b+{ }^7 C_2 a^5 b^2+ { }^7 C_3 a^4 b^3+{ }^7 C_4 a^3 b^4 +{ }^7 C_5 a^2 b^5+{ }^7 C_6 a b^6+b^7$

$\biggr[ { }^7 C_2 = \frac{7!}{2! \cdot 5!} = 21 \biggr]$

$(a+b)^n ={}^nC_0 a^n+{ }^n C_1 a^{n-1} b+{ }^n C_2 a^{n-2} b^2$

$+{ }^n C_3 a^{n-3} b^3+\cdots \cdot{ }^n C_{n-1} a b^{n-1}+ { }^n C_n b^n$

$\quad =\sum_{k=0}^n{ }^n C_k a^{n-k} b^k$

Example $(a-b)^7$

$(a-b)^7= a^7-7 a^6 b+21 a^5 b^2-35 a^4 b^3 +35 a^3 b^4-21 a^2 b^5+7 a b^6-b^7$

$6$ % p.a.$\quad P \rightarrow 1.06 P \rightarrow 1.06 \times 1.06 P$

$(1.06)^{20} \mathrm{P}$

Solution: $(1+0.06)^{20}=1 + 20 \times 0.06+{ }^{20} C_2 0.06^2 +{ }^{20} C_3 0.06^3+{ }^{20} C_4 0.06^4+\ldots$

$\biggl [ { }^{20} C_2=\frac{20 \times 19}{2}=190$

${ }^{20} C_3=\frac{20 \times 19 \times 18}{2 \times 3}=190 \times 6=1140$

${ }^{20} C_4={ }^{20} C_3 \times \frac{17}{4}$

${ }^{20} C_5={ }^{20} C_4 \times \frac{16}{5}\biggl]$

$1 + 1.2 + 190 \times 0.06 \times 0.06 + 190 \times 6 \times 0.06\times 0.06\times 0.06 + 1140 \times \frac{17}{4} \times (0.06)^4 + ….$

$= 1 + 1.2 + 190 \times 0.0036 + 190 \times 6 \times 0.000216 + 1140 \times \frac{17}{4} \times (0.06)^4 + ….$

$\approx 1 + 1.2 + 0.7 + 0.22 + 0.02 + ….$

1. Which is greater? $(1.01)^{1000} \text { or } 11$

Solution:

$1+1000 \times 0.01+{}^{1000} C_2 \times 0.01^2+{ }^{1000} C_3 \times 0.01^3+\cdots$

$1+10+{}^{1000} C_2 \times 0.01^2+{ }^{1000} C_3 \times 0.01^3+\cdots$

2. Evaluate $(1-2 x)^5.$

Solution: $(1-2 x)^5=1-5 \times 2 x+10 \times 4 x^2-10 \times 8 x^3$

$+ 5 \times (2x)^4 - (2x)^5$

$=1-10 x+40 x^2-80 x^3+80 x^4-32 x^5$

3. Evaluate $(x+\frac{1}{x})^6 .$

Solution:

$(x+\frac{1}{x})^6=x^6+6 x^4+15 x^2+20+\frac{15}{{x}^{2}}$

$+ \frac{6}{{x}^{4}}+ \frac{1}{{x}^{6}}$

$\quad$

4. Evaluate $(\frac{2}{x} -\frac{x}{2})^5$

$(\frac{2}{x} -\frac{x}{2})^5=(\frac{2}{x})^5-5(\frac{2}{x})^3+10(\frac{2}{x})-10(\frac{x}{2})$

$+5(\frac{x}{2})^3-(\frac{x}{2})^5$