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If the sum of the coefficients in the expansion of (x+y)n is 1024, then the value of the greatest coefficient in the the expansion is
- a)356
- b)252
- c)210
- d)120
Correct answer is option 'B'. Can you explain this answer?
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If the sum of the coefficients in the expansion of (x+y)n is 1024, the...
Solution:
General term of (x+y)n is given as nCr xn-r yr where r varies from 0 to n.
The sum of the coefficients in the expansion of (x y)n is 1024.
∑nCr = 1024
This is the sum of all the coefficients (including the constant term) in the expansion of (x y)n.
To find the greatest coefficient, we need to find the coefficient with the maximum value.
Coefficient with maximum value:
We know that the coefficients in the expansion of (x y)n are symmetric, i.e., the coefficient of xk in the expansion is equal to the coefficient of yk.
Therefore, the coefficient with maximum value will occur when the power of x and y are equal.
Let this power be r.
So, nCr xr yr = nCr xr xr = nCr x2r
We need to find the value of r such that nCr is maximum.
For this, we need to find the value of r for which nCr is maximum, subject to the condition that the sum of the coefficients is 1024.
We know that the sum of the coefficients is equal to 2n.
2n = 1024
n = 10
So, the expansion of (x y)10 has 11 terms.
nCr xr yr = nCr x2r
For n = 10, we have:
10Cr xr yr = 10Cr x2r
r = 5
Therefore, the coefficient with maximum value is 10C5.
10C5 = 252
Hence, the correct option is (b) 252.
General term of (x+y)n is given as nCr xn-r yr where r varies from 0 to n.
The sum of the coefficients in the expansion of (x y)n is 1024.
∑nCr = 1024
This is the sum of all the coefficients (including the constant term) in the expansion of (x y)n.
To find the greatest coefficient, we need to find the coefficient with the maximum value.
Coefficient with maximum value:
We know that the coefficients in the expansion of (x y)n are symmetric, i.e., the coefficient of xk in the expansion is equal to the coefficient of yk.
Therefore, the coefficient with maximum value will occur when the power of x and y are equal.
Let this power be r.
So, nCr xr yr = nCr xr xr = nCr x2r
We need to find the value of r such that nCr is maximum.
For this, we need to find the value of r for which nCr is maximum, subject to the condition that the sum of the coefficients is 1024.
We know that the sum of the coefficients is equal to 2n.
2n = 1024
n = 10
So, the expansion of (x y)10 has 11 terms.
nCr xr yr = nCr x2r
For n = 10, we have:
10Cr xr yr = 10Cr x2r
r = 5
Therefore, the coefficient with maximum value is 10C5.
10C5 = 252
Hence, the correct option is (b) 252.
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If the sum of the coefficients in the expansion of (x+y)n is 1024, then the value of the greatest coefficient in the the expansion isa)356b)252c)210d)120Correct answer is option 'B'. Can you explain this answer?
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