
The sum of the coefficients of the expression is equal to
[a] 1024
[b] 729
[c] 243
[d] 512
[e] 64
Answer
140.1k+ views
Hint: Expand the given expression by using binomial theorem and find the coefficients of the terms involved in the expression. Calculate the sum of these coefficients to get the result. Instead of making use of binomial theorem, you can also use Pascal's triangle to get the binomial coefficients.
Complete step-by-step answer:
We know from binomial theorem;
Putting n = 6, and in the above expression, we get
Now we know that
Using the above formula, we get
We know that
Using we get
Hence we have
Simplifying, we get
Hence the sum of coefficients = 1+12+60+160+240+192+64=729
Note: [1] Alternative solution 1: Construct pascal triangle till n = 6
Hence we have the binomial coefficients as 1, 6, 15, 20, 15, 6 and 1, which is the same as above.
[2] Alternative solution 2: Best Method.
Let the expansion of the given expression be
Hence we have
Put x = 1, we get
Hence the sum of the coefficients = 729.
[3] Some times, the question asks to find the value of the constant term. In that case, if is within the domain of the expression then put x = 0 to get the result, e.g. Find the constant term in the expansion of the expression .
Since x = 0 is in the domain put x = 0 we get
Hence the constant term in the expansion of the expression is .
Complete step-by-step answer:
We know from binomial theorem;
Putting n = 6,
Now we know that
Using the above formula, we get
We know that
Using we get
Hence we have
Simplifying, we get
Hence the sum of coefficients = 1+12+60+160+240+192+64=729
Note: [1] Alternative solution 1: Construct pascal triangle till n = 6
Hence we have the binomial coefficients as 1, 6, 15, 20, 15, 6 and 1, which is the same as above.
[2] Alternative solution 2: Best Method.
Let the expansion of the given expression be
Hence we have
Put x = 1, we get
Hence the sum of the coefficients = 729.
[3] Some times, the question asks to find the value of the constant term. In that case, if
Since x = 0 is in the domain put x = 0 we get
Hence the constant term in the expansion of the expression is
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