# Pascal's triangle

### Pascal's triangle

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

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