Pascal's triangle

Lessons

• 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 row
  a)

  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)

  ${}_5^{}{C_0}$ + ${}_5^{}{C_1}$ + ${}_5^{}{C_2}$ + ${}_5^{}{C_3}$ + ${}_5^{}{C_4}$
  
  

---