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Question: Sum of the coefficients in the expansion of \[{(x + 2y + z)^{10}}\] A) \[{2^{10}}\] B) \[{3^{10}...

Sum of the coefficients in the expansion of (x+2y+z)10{(x + 2y + z)^{10}}
A) 210{2^{10}}
B) 310{3^{10}}
C) 49{4^9}
D) 410{4^{10}}

Explanation

Solution

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{(x + 2y + z)^{10}}
The expansion (a+b)2{\left( {a + b} \right)^2} is expanded as:-
(a+b)2=a2+b2+2ab{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab
Here the sum of coefficients is =1+1+2 = 1 + 1 + 2
=4= 4
=(1+1)2= {\left( {1 + 1} \right)^2}
Similarly if we put x=1,y=1,z=1x = 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=1x = 1,y = 1,z = 1 we get:-
=(1+2+1)10= {\left( {1 + 2 + 1} \right)^{10}}
Evaluating it further we get:-
=410= {4^{10}}
Hence, the sum of all the coefficients in the expansion of a given expression is 410{4^{10}}.

Hence, option D is the correct option.

Additional information: -
The general binomial expansion of (a+b)n{\left( {a + b} \right)^n} is given by:-
(a+b)n=nC0(a)0(b)n+nC1(a)1(b)n−1+............+nCn(a)n(b)0{\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, (1+x)n=nC0(1)0(x)n+nC1(1)1(x)n−1+............+nCn(1)n(x)0{\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 =2n = {2^n}
Also, the sum of all even coefficients is equal to =2n−1 = {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.