Expand Using the Binomial Theorem (x+y)^20

${(x + y)}^{20}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{20} \frac{20!}{(20 - k)! k!} \cdot {(x)}^{20 - k} \cdot {(y)}^{k}$

Step 2

Expand the summation.

$\frac{20!}{(20 - 0)! 0!} {(x)}^{20 - 0} \cdot {(y)}^{0} + \frac{20!}{(20 - 1)! 1!} {(x)}^{20 - 1} \cdot {(y)}^{1} + \dots + \frac{20!}{(20 - 20)! 20!} {(x)}^{20 - 20} \cdot {(y)}^{20}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{20} \cdot {(y)}^{0} + 20 \cdot {(x)}^{19} \cdot {(y)}^{1} + \dots + 1 \cdot {(x)}^{0} \cdot {(y)}^{20}$

Step 4

Simplify the polynomial result.

$x^{20} + 20 x^{19} y + 190 x^{18} y^{2} + 1140 x^{17} y^{3} + 4845 x^{16} y^{4} + 15504 x^{15} y^{5} + 38760 x^{14} y^{6} + 77520 x^{13} y^{7} + 125970 x^{12} y^{8} + 167960 x^{11} y^{9} + 184756 x^{10} y^{10} + 167960 x^{9} y^{11} + 125970 x^{8} y^{12} + 77520 x^{7} y^{13} + 38760 x^{6} y^{14} + 15504 x^{5} y^{15} + 4845 x^{4} y^{16} + 1140 x^{3} y^{17} + 190 x^{2} y^{18} + 20 x y^{19} + y^{20}$

Expand Using the Binomial Theorem (x+y)^20

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