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) (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} =

    b)
    Investigating Pascal’s Triangle

    c)
    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.
    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 (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)n1{\left( {a + b} \right)^{n - 1}}
    n+1 (a+b)n{\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 (a+b)50{\left( {a + b} \right)^{50}} ?

    d)
    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} + … + nCn2{}_n^{}{C_{n - 2}} + nCn1{}_n^{}{C_{n - 1}} + nCn{}_n^{}{C_n} = 2n{2^n}


  • 3.
    Without using a calculator, evaluate:
    a)
    10C0{}_{10}^{}{C_0} + 10C1{}_{10}^{}{C_1} + 10C2{}_{10}^{}{C_2} + … + 10C9{}_{10}^{}{C_9} + 10C10{}_{10}^{}{C_{10}}

    b)
    5C0{}_5^{}{C_0} + 5C1{}_5^{}{C_1} + 5C2{}_5^{}{C_2} + 5C3{}_5^{}{C_3} + 5C4{}_5^{}{C_4}