Pascal's triangle

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Examples
Lessons
  1. Expand:
    1. i) (a+b)0{\left( {a + b} \right)^0} =
      ii) (a+b)1{\left( {a + b} \right)^1} =
      iii) (a+b)2{\left( {a + b} \right)^2} =
      iv) (a+b)3{\left( {a + b} \right)^3} =

      Use Pascal's Triangle to expand:
      i) (a+b)4{\left( {a + b} \right)^4} =
      ii) (a+b)5{\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 (a+b)0{\left( {a + b} \right)^0}
      2 1 1 (a+b)1{\left( {a + b} \right)^1}
      3 1 2 1 (a+b)2{\left( {a + b} \right)^2}
      4 1 3 3 1 (a+b)3{\left( {a + b} \right)^3}
      5 1 4 6 4 1 (a+b)4{\left( {a + b} \right)^4}
      : : : : :
      n (a+b)n−1{\left( {a + b} \right)^{n - 1}}
      n+1 (a+b)n{\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 (a+b)50{\left( {a + b} \right)^{50}} ?
    4. Express the number pattern of Pascal's triangle in "combination" form, then deduce the following formula:
        \; nC0{}_n^{}{C_0} + nC1{}_n^{}{C_1} + nC2{}_n^{}{C_2} + nC3{}_n^{}{C_3} + … + nCn−2{}_n^{}{C_{n - 2}} + nCn−1{}_n^{}{C_{n - 1}} + nCn{}_n^{}{C_n} = 2n{2^n}
  3. Without using a calculator, evaluate:
    1. 10C0{}_{10}^{}{C_0} + 10C1{}_{10}^{}{C_1} + 10C2{}_{10}^{}{C_2} + … + 10C9{}_{10}^{}{C_9} + 10C10{}_{10}^{}{C_{10}}
    2. 5C0{}_5^{}{C_0} + 5C1{}_5^{}{C_1} + 5C2{}_5^{}{C_2} + 5C3{}_5^{}{C_3} + 5C4{}_5^{}{C_4}
Topic Notes
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A Pascal's triangle is a number triangle of the binomial coefficients. The first row of the triangle is always 1.
Basic Concepts
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