Pascal's triangle

Lessons

1. 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
2. Pascal's Triangle - sum of numbers in each row

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

   2. What is the sum of the numbers in the 10th row of Pascal's Triangle?
   3. What is the sum of the coefficients in the expansion of ${\left( {a + b} \right)^{50}}$ ?
   4. 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:
   1. ${}_{10}^{}{C_0}$ + ${}_{10}^{}{C_1}$ + ${}_{10}^{}{C_2}$ + … + ${}_{10}^{}{C_9}$ + ${}_{10}^{}{C_{10}}$
   2. ${}_5^{}{C_0}$ + ${}_5^{}{C_1}$ + ${}_5^{}{C_2}$ + ${}_5^{}{C_3}$ + ${}_5^{}{C_4}$