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- Binomial Expansion

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Get Started Now- Lesson: 1a54:32
- Lesson: 1b10:52
- Lesson: 2a5:28
- Lesson: 2b1:03
- Lesson: 2c1:10
- Lesson: 2d10:06
- Lesson: 3a2:33
- Lesson: 3b6:26

A Pascal's triangle is a number triangle of the binomial coefficients. The first row of the triangle is always 1.

- 1.Expand:a)i) ${\left( {a + b} \right)^0}$ =

ii) ${\left( {a + b} \right)^1}$ =

iii) ${\left( {a + b} \right)^2}$ =

iv) ${\left( {a + b} \right)^3}$ =

Use Pascal's Triangle to expand:

i) ${\left( {a + b} \right)^4}$ =

ii) ${\left( {a + b} \right)^5}$ =b)Investigating Pascal's Triangle - 2.Pascal's Triangle - sum of numbers in each rowa)
Row Pattern Corresponding binomial expression Sum of the numbers in,the row Express the sum as a power of 2 1 1 ${\left( {a + b} \right)^0}$ 2 1 1 ${\left( {a + b} \right)^1}$ 3 1 2 1 ${\left( {a + b} \right)^2}$ 4 1 3 3 1 ${\left( {a + b} \right)^3}$ 5 1 4 6 4 1 ${\left( {a + b} \right)^4}$ : : : : : n ${\left( {a + b} \right)^{n - 1}}$ n+1 ${\left( {a + b} \right)^n}$ b)What is the sum of the numbers in the 10th row of Pascal's Triangle?c)What is the sum of the coefficients in the expansion of ${\left( {a + b} \right)^{50}}$ ?d)Express the number pattern of Pascal's triangle in "combination" form, then deduce the following formula:

$\;$ ${}_n^{}{C_0}$ + ${}_n^{}{C_1}$ + ${}_n^{}{C_2}$ + ${}_n^{}{C_3}$ + … + ${}_n^{}{C_{n - 2}}$ + ${}_n^{}{C_{n - 1}}$ + ${}_n^{}{C_n}$ = ${2^n}$ - 3.Without using a calculator, evaluate:a)${}_{10}^{}{C_0}$ + ${}_{10}^{}{C_1}$ + ${}_{10}^{}{C_2}$ + … + ${}_{10}^{}{C_9}$ + ${}_{10}^{}{C_{10}}$b)${}_5^{}{C_0}$ + ${}_5^{}{C_1}$ + ${}_5^{}{C_2}$ + ${}_5^{}{C_3}$ + ${}_5^{}{C_4}$

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