Question

The coefficient of ${{x}^{2017}}$in ${{\sum\nolimits_{r=0}^{2020}{{}^{2020}{{C}_{r}}\left( x-2018 \right)}}^{2020-r}}{{\left( 2017 \right)}^{r}}$ is

Answer 1

Hint:
Here we have to find the coefficient of ${{x}^{2017}}$in the given expression. To find the required coefficient, we will first equate the power of (x-2018) with 2017. From there, we will find the value of r and we will put the value in the given expression, from there we will get the coefficient of ${{x}^{2017}}$ after putting the value of r in the expression.

Complete step by step solution:
Here, we have to find the coefficient of ${{x}^{2017}}$in${{\sum\nolimits_{r=0}^{2020}{{}^{2020}{{C}_{r}}\left( x-2018 \right)}}^{2020-r}}{{\left( 2017 \right)}^{r}}$.
Now, we will equate the exponent of (x-2018) i.e. $2020-r$ with 2017.
$2020-r=2017$
Adding r on both sides, we get
$2020-r+r=2017+r$
On simplifying the terms further, we get
$2020=2017+r$
Now, we will subtract 2017 from 2020.
$
  2020-2017=r \\
  \therefore r=3 \\
$
We will put the value of r in the expression.
Therefore, the expression becomes
${}^{2020}{{C}_{3}}{{\left( x-2018 \right)}^{2020-3}}{{\left( 2017 \right)}^{3}}$
On further simplification, we get
${}^{2020}{{C}_{3}}{{\left( x-2018 \right)}^{2017}}{{\left( 2017 \right)}^{3}}$
We will evaluate the value of $2020{C_3}$
Therefore,
${}^{2020}{{C}_{3}}=\dfrac{2020!}{3!2017!}$
After putting the value in the expression, we get
$\dfrac{2020!}{3!2017!}{{\left( x-2018 \right)}^{2017}}{{\left( 2017 \right)}^{3}}$
Here we need the coefficient of${{x}^{2017}}$, and there will be only one term of ${{x}^{2017}}$in its expansion.
Therefore, the coefficient of ${{x}^{2017}}$is equal to $\dfrac{2020!}{3!2017!}{{\left( 2017 \right)}^{3}}$
This can also be written as $\dfrac{2020!}{3!2016!}{{\left( 2017 \right)}^{2}}$

Hence, the coefficient of ${{x}^{2017}}$is equal to $\dfrac{2020!}{3!2016!}{{\left( 2017 \right)}^{2}}$

Note:
Since we have used factorials here. So we need to know its meaning and definition of factorial.
The definition and some important properties of factorial are as follows:-
1) Factorial of any positive integer is defined as the multiplication of all the positive integers less than or equal to the given positive integers.
2) Factorial of zero is one.
3) Factorials are commonly used in permutations and combinations problems.
4) Factorials of negative integers are not defined.