Examples
Exercise 8B
- a. \(4! = 4\times 3\times 2\times 1 = 24\)
b. \(9! = 9 \times 8 \times 7\times 6\times 5\times 4\times 3\times 2\times 1 = 362,880\)
c. \(\frac{10!}{7!} = 720\)
d. \(\frac{15!}{13!} = 210\)
- a. \(^{4}C_{2} = \frac{4!}{2!\times (4-2!)} =\frac{24}{4} = 6\)
\(\left( 10-\frac{1}{2}x\right)^6\) (first 4 terms)
\[10^6-^6C_{1}(10^5)\left( \frac{1}{2}x \right)+^6C_{2}(10^4)\left( \frac{1}{2}x \right)^2-^6C_{3}(10^3)\left( \frac{1}{2}x \right)^3$$
$$=$$
$$1000000-300000x+37500x^2-2500x^3+\dots\]
Exercise 8C
- a.
\(\((1+x)^{4}=1^4+^4C_{1}(1^3)(x)+^4C_{2}(1^2)(x^2)+^4C_{3}(1^1)(x^3)+x^4\)\)
\(\(=\)\)
\(\(1+4x+6x^2+4x^3+x^4\)\)
b.
\(\((3+x)^{4}=3^4+^4C_{1}(3^3)(x)+^4C_{2}(3^2)(x^2)+^4C_{3}(3^1)(x^3)+x^4\)\)
\(\(=\)\)
\(\(81+108x+54x^2+12x^3+x^4\)\)
Only first 4 terms: - a.
\(\((1+x)^{10} = 1^{10}+^{10}C_{1}(1^9)(x)+^{10}C_{2}(1^8)(x^2)+^{10}C_{3}(1^7)(x^3)\)\)
\(\(=\)\)
\(\(1+10x+45x^2+120x^3\)\)
b.
\(\((1-2x)^{5}=1^5+^5C_{1}(1^4)(-2x)+^5C_{2}(1^3)(-2^3)\)\)