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# Binomial Theorem Calculator

## Get detailed solutions to your math problems with our Binomial Theorem step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Here, we show you a step-by-step solved example of binomial theorem. This solution was automatically generated by our smart calculator:

$\left(x+3\right)^5$
2

We can expand the expression $\left(x+3\right)^5$ using Newton's binomial theorem, which is a formula that allow us to find the expanded form of a binomial raised to a positive integer $n$. The formula is as follows: $\displaystyle(a\pm b)^n=\sum_{k=0}^{n}\left(\begin{matrix}n\\k\end{matrix}\right)a^{n-k}b^k=\left(\begin{matrix}n\\0\end{matrix}\right)a^n\pm\left(\begin{matrix}n\\1\end{matrix}\right)a^{n-1}b+\left(\begin{matrix}n\\2\end{matrix}\right)a^{n-2}b^2\pm\dots\pm\left(\begin{matrix}n\\n\end{matrix}\right)b^n$. The number of terms resulting from the expansion always equals $n + 1$. The coefficients $\left(\begin{matrix}n\\k\end{matrix}\right)$ are combinatorial numbers which correspond to the nth row of the Tartaglia triangle (or Pascal's triangle). In the formula, we can observe that the exponent of $a$ decreases, from $n$ to $0$, while the exponent of $b$ increases, from $0$ to $n$. If one of the binomial terms is negative, the positive and negative signs alternate.

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 3^{0}x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3^{1}x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
3

Calculate the power $3^{0}$

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3^{1}x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
4

Calculate the power $3^{1}$

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
5

Calculate the power $3^{2}$

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
6

Calculate the power $3^{3}$

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
7

Calculate the power $3^{4}$

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
8

Calculate the power $3^{5}$

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
9

Any expression to the power of $1$ is equal to that same expression

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
10

Any expression multiplied by $1$ is equal to itself

$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
11

Any expression (except $0$ and $\infty$) to the power of $0$ is equal to $1$

$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
12

Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
13

The factorial of $0$ is

$\frac{5!}{1\cdot 1}x^{5}$
14

The factorial of $5$ is

$\frac{120}{1\cdot 1}x^{5}$
15

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}x^{5}$
16

Any expression divided by one ($1$) is equal to that same expression

$120x^{5}$
17

Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
18

The factorial of $0$ is

$\frac{5!}{1\cdot 1}x^{5}$
19

The factorial of $5$ is

$\frac{120}{1\cdot 1}x^{5}$
20

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}x^{5}$
21

Any expression divided by one ($1$) is equal to that same expression

$120x^{5}$
22

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
23

The factorial of $1$ is

$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
24

The factorial of $5$ is

$\frac{120}{1\cdot 1}\cdot 3x^{4}$
25

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}\cdot 3x^{4}$
26

Any expression divided by one ($1$) is equal to that same expression

$120\cdot 3x^{4}$
27

Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
28

The factorial of $0$ is

$\frac{5!}{1\cdot 1}x^{5}$
29

The factorial of $5$ is

$\frac{120}{1\cdot 1}x^{5}$
30

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}x^{5}$
31

Any expression divided by one ($1$) is equal to that same expression

$120x^{5}$
32

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
33

The factorial of $1$ is

$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
34

The factorial of $5$ is

$\frac{120}{1\cdot 1}\cdot 3x^{4}$
35

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}\cdot 3x^{4}$
36

Any expression divided by one ($1$) is equal to that same expression

$120\cdot 3x^{4}$
37

Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
38

The factorial of $2$ is

$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
39

The factorial of $5$ is

$\frac{120}{2\cdot 1}\cdot 9x^{3}$
40

Any expression multiplied by $1$ is equal to itself

$\frac{120}{2}\cdot 9x^{3}$
41

Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
42

The factorial of $0$ is

$\frac{5!}{1\cdot 1}x^{5}$
43

The factorial of $5$ is

$\frac{120}{1\cdot 1}x^{5}$
44

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}x^{5}$
45

Any expression divided by one ($1$) is equal to that same expression

$120x^{5}$
46

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
47

The factorial of $1$ is

$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
48

The factorial of $5$ is

$\frac{120}{1\cdot 1}\cdot 3x^{4}$
49

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}\cdot 3x^{4}$
50

Any expression divided by one ($1$) is equal to that same expression

$120\cdot 3x^{4}$
51

Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
52

The factorial of $2$ is

$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
53

The factorial of $5$ is

$\frac{120}{2\cdot 1}\cdot 9x^{3}$
54

Any expression multiplied by $1$ is equal to itself

$\frac{120}{2}\cdot 9x^{3}$
55

Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
56

The factorial of $3$ is

$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
57

The factorial of $5$ is

$\frac{120}{6\cdot 1}\cdot 27x^{2}$
58

Any expression multiplied by $1$ is equal to itself

$\frac{120}{6}\cdot 27x^{2}$
59

Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
60

The factorial of $0$ is

$\frac{5!}{1\cdot 1}x^{5}$
61

The factorial of $5$ is

$\frac{120}{1\cdot 1}x^{5}$
62

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}x^{5}$
63

Any expression divided by one ($1$) is equal to that same expression

$120x^{5}$
64

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
65

The factorial of $1$ is

$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
66

The factorial of $5$ is

$\frac{120}{1\cdot 1}\cdot 3x^{4}$
67

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}\cdot 3x^{4}$
68

Any expression divided by one ($1$) is equal to that same expression

$120\cdot 3x^{4}$
69

Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
70

The factorial of $2$ is

$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
71

The factorial of $5$ is

$\frac{120}{2\cdot 1}\cdot 9x^{3}$
72

Any expression multiplied by $1$ is equal to itself

$\frac{120}{2}\cdot 9x^{3}$
73

Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
74

The factorial of $3$ is

$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
75

The factorial of $5$ is

$\frac{120}{6\cdot 1}\cdot 27x^{2}$
76

Any expression multiplied by $1$ is equal to itself

$\frac{120}{6}\cdot 27x^{2}$
77

Calculate the binomial coefficient $\left(\begin{matrix}5\\4\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(4!\right)\left(5-4\right)!}\cdot 81x$
78

The factorial of $4$ is

$\frac{5!}{24\cdot 1}\cdot 81x$
79

The factorial of $5$ is

$\frac{120}{24\cdot 1}\cdot 81x$
80

Any expression multiplied by $1$ is equal to itself

$\frac{120}{24}\cdot 81x$
81

Subtract the values $5$ and $-1$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(5-2\right)!}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(5-3\right)!}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
82

Subtract the values $5$ and $-2$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(5-3\right)!}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
83

Subtract the values $5$ and $-3$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
84

Subtract the values $5$ and $-4$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
85

Add the values $5$ and $0$

$\frac{5!}{\left(0!\right)\left(5!\right)}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
86

Simplify the fraction

$\frac{1}{0!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
87

Multiply the fraction by the term

$\frac{1x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
88

Any expression multiplied by $1$ is equal to itself

$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
89

Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
90

The factorial of $0$ is

$\frac{5!}{1\cdot 1}x^{5}$
91

The factorial of $5$ is

$\frac{120}{1\cdot 1}x^{5}$
92

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}x^{5}$
93

Any expression divided by one ($1$) is equal to that same expression

$120x^{5}$
94

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
95

The factorial of $1$ is

$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
96

The factorial of $5$ is

$\frac{120}{1\cdot 1}\cdot 3x^{4}$
97

Any expression multiplied by $1$ is equal to itself

$\frac{120}{1}\cdot 3x^{4}$
98

Any expression divided by one ($1$) is equal to that same expression

$120\cdot 3x^{4}$
99

Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
100

The factorial of $2$ is

$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
101

The factorial of $5$ is

$\frac{120}{2\cdot 1}\cdot 9x^{3}$
102

Any expression multiplied by $1$ is equal to itself

$\frac{120}{2}\cdot 9x^{3}$
103

Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
104

The factorial of $3$ is

$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
105

The factorial of $5$ is

$\frac{120}{6\cdot 1}\cdot 27x^{2}$
106

Any expression multiplied by $1$ is equal to itself

$\frac{120}{6}\cdot 27x^{2}$
107

Calculate the binomial coefficient $\left(\begin{matrix}5\\4\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\frac{5!}{\left(4!\right)\left(5-4\right)!}\cdot 81x$
108

The factorial of $4$ is

$\frac{5!}{24\cdot 1}\cdot 81x$
109

The factorial of $5$ is

$\frac{120}{24\cdot 1}\cdot 81x$
110

Any expression multiplied by $1$ is equal to itself

$\frac{120}{24}\cdot 81x$
111

Calculate the binomial coefficient $\left(\begin{matrix}5\\5\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

$\left(\frac{5!}{\left(5!\right)\left(5-5\right)!}\right)\cdot 243$
112

Simplify the fraction

$\left(\frac{1}{\left(5-5\right)!}\right)\cdot 243$
113

Subtract the values $5$ and $-5$

$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243\left(5!\right)}{\left(5!\right)\left(0!\right)}$
114

Simplify the fraction

$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
115

The factorial of $0$ is

$\frac{x^{5}}{1}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
116

The factorial of $1$ is

$\frac{x^{5}}{1}+\frac{3\left(5!\right)}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
117

The factorial of $5$ is

$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
118

The factorial of $2$ is

$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
119

The factorial of $5$ is

$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
120

Multiply $1$ times $24$

$\frac{x^{5}}{1}+\frac{3\cdot 120}{24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
121

Multiply $3$ times $120$

$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
122

Multiply $2$ times $6$

$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{9\cdot 120}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
123

Multiply $9$ times $120$

$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{1080}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
124

Divide $360$ by $24$

$\frac{x^{5}}{1}+15x^{4}+\frac{1080}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
125

Divide $1080$ by $12$

$\frac{x^{5}}{1}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
126

Any expression divided by one ($1$) is equal to that same expression

$x^{5}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
127

The factorial of $3$ is

$x^{5}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
128

The factorial of $5$ is

$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
129

The factorial of $4$ is

$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{24\cdot 1}x+\frac{243}{0!}$
130

The factorial of $5$ is

$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{0!}$
131

The factorial of $0$ is

$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
132

Multiply $6$ times $2$

$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{12}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
133

Multiply $27$ times $120$

$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
134

Multiply $24$ times $1$

$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{81\cdot 120}{24}x+\frac{243}{1}$
135

Multiply $81$ times $120$

$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{9720}{24}x+\frac{243}{1}$
136

Divide $3240$ by $12$

$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{9720}{24}x+\frac{243}{1}$
137

Divide $9720$ by $24$

$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+\frac{243}{1}$
138

Divide $243$ by $1$

$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$

##  Final answer to the problem

$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$

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