Question

The coefficient of \[{x^{1012}}\] in the expansion of \[{\left( {1 + {x^n} + {x^{253}}} \right)^{10}}\],(where \[n \leqslant 22\] is any positive integer),is:
A) 1
B) \[^{10}{C_4}\]
C) 4n
D) \[^{253}{C_4}\]

Answer 1

Hint: Here first we will assume the given expression to be P and then write its general term using the following formula:-
If the expression is of the form \[{\left( {a + b} \right)^n}\] then its general term is:-
\[{T_{r + 1}}{ = ^n}{C_r}{\left( a \right)^{n - r}}{\left( b \right)^r}\] and then we will compute the value of r by comparing and find the coefficient of \[{x^{1012}}\].

Complete step-by-step answer:
Let us assume
\[P = {\left( {1 + {x^n} + {x^{253}}} \right)^{10}}\]
It can be written as:-
\[P = {\left( {\left( {1 + {x^n}} \right) + {x^{253}}} \right)^{10}}\]
Now let
 \[
  a = 1 + {x^n} \\
  b = {x^{253}} \\
 \]
And \[n = 10\]
Now we will apply the following formula to find the general term of the given expression:-
\[{T_{r + 1}}{ = ^n}{C_r}{\left( a \right)^{n - r}}{\left( b \right)^r}\]
Putting in the respective values we get:-
\[{T_{r + 1}}{ = ^{10}}{C_r}{\left( {1 + {x^n}} \right)^{10 - r}}{\left( {{x^{253}}} \right)^r}\]
On simplifying we get:-
\[{T_{r + 1}}{ = ^{10}}{C_r}{\left( {1 + {x^n}} \right)^{10 - r}}\left( {{x^{253r}}} \right)\]……………………………(1)
Now since we need to find the coefficient of \[{x^{1012}}\]
Therefore we will equate the power of x in the above equation with 1012 in order to get the value of r
Hence on equating we get:-
\[253r = 1012\]
Solving for r we get:-
\[
  r = \dfrac{{1012}}{{253}} \\
   \Rightarrow r = 4 \\
 \]
Putting this value in equation 1 we get:-
\[{T_{r + 1}}{ = ^{10}}{C_4}{\left( {1 + {x^n}} \right)^{10 - r}}\left( {{x^{253\left( 4 \right)}}} \right)\]
Simplifying it we get:-
\[{T_{r + 1}}{ = ^{10}}{C_4}{\left( {1 + {x^n}} \right)^{10 - r}}\left( {{x^{1012}}} \right)\]
Hence the coefficient of \[{x^{1012}}\] is \[^{10}{C_4}\]

Therefore option B is the correct option.

Note: Students should note that coefficient is the constant term with which a variable term is multiplied.
Also, students should keep the formula for the general term of a binomial term in mind in order to get the correct answer.
The general binomial expansion of \[{\left( {a + b} \right)^n}\] is given by:-
\[{\left( {a + b} \right)^n}{ = ^n}{C_0}{\left( a \right)^0}{\left( b \right)^n}{ + ^n}{C_1}{\left( a \right)^1}{\left( b \right)^{n - 1}}{ + ^n}{C_3}{\left( a \right)^3}{\left( b \right)^{n - 3}}.................................{ + ^n}{C_n}{\left( a \right)^n}{\left( b \right)^0}\]