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Question

What number does C3 represent in the expansion C subscript 0 a to the power of 5 plus C subscript 1 a to the power of 4 b plus C subscript 2 a squared b squared plus C subscript 3 a b cubed plus C subscript 4 b to the power of 4 ? Explain.

The correct answer is: = 4


    ANSWER:
    Hint:
    The binomial expansion is  , left parenthesis x plus y right parenthesis to the power of n equals sum from k equals 0 to n of   n C subscript k x to the power of n minus k end exponent y to the power of k text , here  end text n greater or equal than 0.
    The value of n C subscript r equals fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction
    We are asked to find what number does C subscript 3 represent in the expansion .
    C subscript 0 a to the power of 5 plus C subscript 1 a to the power of 4 b plus C subscript 2 a squared b squared plus C subscript 3 a b cubed plus C subscript 4 b to the power of 4.
    Step 1 of 2:
    The expansion is: C subscript 0 a to the power of 5 plus C subscript 1 a to the power of 4 b plus C subscript 2 a squared b squared plus C subscript 3 a b cubed plus C subscript 4 b to the power of 4
    Here, C subscript 3 represents the coefficient of in the expansion.
    Step 2 of 2:
    The value of n=4, so the value of C subscript 3 is:
    4 C subscript 3 equals fraction numerator 4 factorial over denominator 3 factorial left parenthesis 4 minus 3 right parenthesis factorial end fraction
    equals fraction numerator 4 factorial over denominator 3 factorial cross times 1 factorial end fraction
    = 4
    Note:
    The value n C subscript r equals fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction is called the combination permutation.

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