Question

Sum of the coefficients in the expansion of \[{(x + 2y + z)^{10}}\]
A) \[{2^{10}}\]
B) \[{3^{10}}\]
C) \[{4^9}\]
D) \[{4^{10}}\]

Answer 1

Hint: Here we will just put each of the variables equal to 1 and then evaluate the value of the given expression to get the sum of all coefficients.

Complete step by step answer:
The given expression is \[{(x + 2y + z)^{10}}\]
The expansion \[{\left( {a + b} \right)^2}\] is expanded as:-
\[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\]
Here the sum of coefficients is \[ = 1 + 1 + 2\]
\[ = 4\]
\[ = {\left( {1 + 1} \right)^2}\]
Similarly if we put \[x = 1,y = 1,z = 1\] in the given expression and then evaluate the value of the expression, we will get the sum of all the coefficients in its expansion.
Hence on putting \[x = 1,y = 1,z = 1\] we get:-
\[ = {\left( {1 + 2 + 1} \right)^{10}}\]
Evaluating it further we get:-
\[ = {4^{10}}\]
Hence, the sum of all the coefficients in the expansion of a given expression is \[{4^{10}}\].

Hence, option D is the correct option.

Additional information: -
The general binomial expansion of \[{\left( {a + b} \right)^n}\] is given by:-
\[{\left( {a + b} \right)^n} = {}^n{C_0}{\left( a \right)^0}{\left( b \right)^n}{ + ^n}{C_1}{\left( a \right)^1}{\left( b \right)^{n - 1}} + ............{ + ^n}{C_n}{\left( a \right)^n}{\left( b \right)^0}\]
Also, \[{\left( {1 + x} \right)^n} = {}^n{C_0}{\left( 1 \right)^0}{\left( x \right)^n}{ + ^n}{C_1}{\left( 1 \right)^1}{\left( x \right)^{n - 1}} + ............{ + ^n}{C_n}{\left( 1 \right)^n}{\left( x \right)^0}\]
The sum of all the coefficients in this expansion is \[ = {2^n}\]
Also, the sum of all even coefficients is equal to \[ = {2^{n - 1}}\]

Note:
Students should take a note that in such questions where we have to find the sum of the coefficients we need to put each of the variables equal to 1 and evaluate the value to make the calculations easier.