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The coefficient of x7 in the expansion of (1– x – x2 + x3 )6 is [2011]
- a)–132
- b)–144
- c)132
- d)144
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
The coefficient of x7 in the expansion of (1– x – x2 + x3 ...
(1 – x – x2 + x3)6 = [(1– x) – x2 (1 – x)]6
= (1– x)6 (1 – x2)6
= (1 – 6x + 15x2 – 20x3 + 15x4 – 6x5 + x6)
× (1 – 6x2 + 15x4 – 20x6 + 15x8 – 6x10 + x12)
Coefficient of x7 = (– 6) (– 20) + (– 20)(15) + (– 6) (–6)
= – 144
= (1– x)6 (1 – x2)6
= (1 – 6x + 15x2 – 20x3 + 15x4 – 6x5 + x6)
× (1 – 6x2 + 15x4 – 20x6 + 15x8 – 6x10 + x12)
Coefficient of x7 = (– 6) (– 20) + (– 20)(15) + (– 6) (–6)
= – 144
Most Upvoted Answer
The coefficient of x7 in the expansion of (1– x – x2 + x3 ...
The coefficient of x^7 in the expansion of (1 + x)^n can be found using the binomial theorem. According to the theorem, the coefficient of x^k in the expansion of (1 + x)^n is given by the binomial coefficient C(n, k), where n is the power to which (1 + x) is raised and k is the power of x in the term.
In this case, we are looking for the coefficient of x^7, so k = 7. We need to determine the value of n.
The binomial theorem states that (1 + x)^n = C(n, 0) + C(n, 1)x + C(n, 2)x^2 + ... + C(n, n)x^n.
Since we are looking for the coefficient of x^7, we can set k = 7 and solve for n:
C(n, 7) = C(n, 7)
This means that the coefficient of x^7 is C(n, 7).
Therefore, the coefficient of x^7 in the expansion of (1 + x)^n is C(n, 7).
In this case, we are looking for the coefficient of x^7, so k = 7. We need to determine the value of n.
The binomial theorem states that (1 + x)^n = C(n, 0) + C(n, 1)x + C(n, 2)x^2 + ... + C(n, n)x^n.
Since we are looking for the coefficient of x^7, we can set k = 7 and solve for n:
C(n, 7) = C(n, 7)
This means that the coefficient of x^7 is C(n, 7).
Therefore, the coefficient of x^7 in the expansion of (1 + x)^n is C(n, 7).
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The coefficient of x7 in the expansion of (1– x – x2 + x3 )6 is [2011]a)–132b)–144c)132d)144Correct answer is option 'B'. Can you explain this answer?
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