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