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Use the binomial theorem to expand the expressions: left parenthesis n plus 5 right parenthesis to the power of 5

hintHint:

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 comma text  here  end text n greater or equal than 0 .
We are asked to find the expansion of left parenthesis n plus 5 right parenthesis to the power of 5 using Binomial theorem.

The correct answer is: (n+1)th row of the triangle.


    Step 1 of 2:
    The given expression is left parenthesis n plus 5 right parenthesis to the power of 5 . Here, the value of n=5, thus we would 5+1=6 terms. The values of x equals n straight & y equals 5
    Step 2 of 2:
    Substitute the values in the binomial equation;

    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell left parenthesis n plus 5 right parenthesis to the power of 5 equals 5 C subscript 0 open parentheses n to the power of 5 close parentheses plus 5 C subscript 1 open parentheses n to the power of 4 close parentheses left parenthesis 5 right parenthesis plus 5 C subscript 2 open parentheses n cubed close parentheses open parentheses 5 squared close parentheses plus 5 C subscript 3 open parentheses n squared close parentheses open parentheses 5 cubed close parentheses plus 5 C subscript 4 left parenthesis n right parenthesis open parentheses 5 to the power of 4 close parentheses plus 5 C subscript 5 open parentheses 5 to the power of 5 close parentheses end cell row cell equals n to the power of 5 plus 5 n to the power of 4 left parenthesis 5 right parenthesis plus 10 n cubed left parenthesis 25 right parenthesis plus 10 n squared left parenthesis 125 right parenthesis plus 5 n left parenthesis 625 right parenthesis plus 3125 end cell row cell equals n to the power of 5 plus 25 n to the power of 4 plus 250 n cubed plus 1250 n squared plus 3125 n plus 3125 end cell end table
    Thus, the expansion is; left parenthesis n plus 5 right parenthesis to the power of 5 equals n to the power of 5 plus 25 n to the power of 4 plus 250 n cubed plus 1250 n squared plus 3125 n plus 3125

    The answer can also be found using the Pascal’s triangle. For the expansion of the expression (x + y)n , we would consider the (n+1)th row in the triangle.

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