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The co-efficient of x4 in the expansion of (1 – x + 2x2)12 is
- a)12C3
- b)13C3
- c)14C4
- d)12C3 + 3 13C3 + 14C4
Correct answer is option 'D'. Can you explain this answer?
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The co-efficient of x4in the expansion of (1 –x + 2x2)12isa)12C3...
Understanding the Expansion
To find the coefficient of x^4 in the expansion of (1 - x + 2x^2)^12, we apply the multinomial theorem.
Multinomial Expansion
The expression (a + b + c)^n can be expanded as:
- Sum over all non-negative integers i, j, k such that i + j + k = n.
- The term is given by n!/(i!j!k!) * a^i * b^j * c^k.
In our case:
- a = 1
- b = -x
- c = 2x^2
- n = 12
Finding Relevant Terms
We need to find combinations of i, j, and k such that the total power of x is 4:
- From b = -x: contributes j to x.
- From c = 2x^2: contributes 2k to x.
The equation to satisfy is:
j + 2k = 4
Also, we have the constraint:
i + j + k = 12
Possible Combinations
Now, we analyze the possible pairs (j, k):
1. k = 0: j = 4, i = 12 - 4 - 0 = 8
2. k = 1: j = 2, i = 12 - 2 - 1 = 9
3. k = 2: j = 0, i = 12 - 0 - 2 = 10
These yield three combinations (i, j, k):
- (8, 4, 0)
- (9, 2, 1)
- (10, 0, 2)
Calculating Coefficients
Now, we compute the contribution of each combination:
1. For (8, 4, 0): Coefficient = 12!/(8!4!0!) = 495
2. For (9, 2, 1): Coefficient = 12!/(9!2!1!) * (-1)^2 * 2^1 = 660
3. For (10, 0, 2): Coefficient = 12!/(10!0!2!) * (2^2) = 66
Final Coefficient
Summing these coefficients gives:
495 + 660 + 66 = 1221
Thus, the coefficient of x^4 in the expansion is:
Correct Answer
Option D: 12C3 + 13C3 + 14C4.
To find the coefficient of x^4 in the expansion of (1 - x + 2x^2)^12, we apply the multinomial theorem.
Multinomial Expansion
The expression (a + b + c)^n can be expanded as:
- Sum over all non-negative integers i, j, k such that i + j + k = n.
- The term is given by n!/(i!j!k!) * a^i * b^j * c^k.
In our case:
- a = 1
- b = -x
- c = 2x^2
- n = 12
Finding Relevant Terms
We need to find combinations of i, j, and k such that the total power of x is 4:
- From b = -x: contributes j to x.
- From c = 2x^2: contributes 2k to x.
The equation to satisfy is:
j + 2k = 4
Also, we have the constraint:
i + j + k = 12
Possible Combinations
Now, we analyze the possible pairs (j, k):
1. k = 0: j = 4, i = 12 - 4 - 0 = 8
2. k = 1: j = 2, i = 12 - 2 - 1 = 9
3. k = 2: j = 0, i = 12 - 0 - 2 = 10
These yield three combinations (i, j, k):
- (8, 4, 0)
- (9, 2, 1)
- (10, 0, 2)
Calculating Coefficients
Now, we compute the contribution of each combination:
1. For (8, 4, 0): Coefficient = 12!/(8!4!0!) = 495
2. For (9, 2, 1): Coefficient = 12!/(9!2!1!) * (-1)^2 * 2^1 = 660
3. For (10, 0, 2): Coefficient = 12!/(10!0!2!) * (2^2) = 66
Final Coefficient
Summing these coefficients gives:
495 + 660 + 66 = 1221
Thus, the coefficient of x^4 in the expansion is:
Correct Answer
Option D: 12C3 + 13C3 + 14C4.
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The co-efficient of x4in the expansion of (1 –x + 2x2)12isa)12C3b)13C3c)14C4d)12C3+ 313C3+14C4Correct answer is option 'D'. Can you explain this answer?
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