Pascal's triangle

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Now Playing:Pascals triangle – Example 1a

Expand:
1. 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) ${\left( {a + b} \right)^5}$ =
2. Investigating Pascal's Triangle

Pascal's Triangle - sum of numbers in each row

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		${\left( {a + b} \right)^n}$

What is the sum of the numbers in the 10th row of Pascal's Triangle?
What is the sum of the coefficients in the expansion of ${\left( {a + b} \right)^{50}}$ ?
Express the number pattern of Pascal's triangle in "combination" form, then deduce the following formula:
$\;$ ${}_n^{}{C_0}$ + ${}_n^{}{C_1}$ + ${}_n^{}{C_2}$ + ${}_n^{}{C_3}$ + … + ${}_n^{}{C_{n - 2}}$ + ${}_n^{}{C_{n - 1}}$ + ${}_n^{}{C_n}$ = ${2^n}$

Without using a calculator, evaluate:
1. ${}_{10}^{}{C_0}$ + ${}_{10}^{}{C_1}$ + ${}_{10}^{}{C_2}$ + … + ${}_{10}^{}{C_9}$ + ${}_{10}^{}{C_{10}}$
2. ${}_5^{}{C_0}$ + ${}_5^{}{C_1}$ + ${}_5^{}{C_2}$ + ${}_5^{}{C_3}$ + ${}_5^{}{C_4}$

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