can you check my answer?
Find the coefficient of the term a6b3 in the binomial expansion of the expression (a - 4b)9.
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5,376
344,064
-5,376
-344,064
answer: b
(9C6)a6(−1)7(4b)3=−5,376a6b3
To find the coefficient of the term a6b3 in the binomial expansion of (a - 4b)9, we can use the binomial theorem.
The formula for the coefficient of a specific term in a binomial expansion is given by:
nCr * a^r * b^(n-r)
where n is the exponent of the binomial, r is the power of a in the term, n-r is the power of b in the term, and nCr represents the binomial coefficient.
In this case, we need to find the coefficient for the term with a power of 6 (a6) and b power of 3 (b3).
Using the formula, we can substitute the values:
The binomial coefficient, 9C6, can be calculated as:
9C6 = 9! / (6! * (9-6)!) = 9! / (6! * 3!) = (9 * 8 * 7) / (3 * 2 * 1) = 84
The exponent of a, 6, means a^6.
The exponent of b, 3, means b^3.
Putting it all together:
Coefficient of a6b3 = 9C6 * a^6 * b^3 = 84 * a^6 * b^3
Therefore, the correct answer is b.
To find the coefficient of the term a^6b^3 in the binomial expansion of (a - 4b)^9, we can use the binomial theorem.
The binomial theorem states that the coefficient of the term a^r b^s in the expansion of (a + b)^n is given by the formula:
C(n, r) * a^(n-r) * b^r
Where C(n, r) represents the binomial coefficient, given by the formula:
C(n, r) = n! / (r! * (n-r)!)
In this case, we need to find the coefficient of the term a^6 b^3 in the expansion of (a - 4b)^9. Hence, we have:
n = 9
r = 6
s = 3
First, calculate the binomial coefficient C(9, 6):
C(9, 6) = 9! / (6! * (9-6)!)
= 9! / (6! * 3!)
= (9 * 8 * 7) / (3 * 2 * 1)
= 84
Next, substitute the values into the formula for the coefficient:
Coefficient = C(9, 6) * a^(9-6) * b^6
= 84 * a^3 * b^6
= 84a^3b^6
Therefore, the coefficient of the term a^6 b^3 in the binomial expansion of (a - 4b)^9 is 84.
Comparing this with the given answer choices, the correct answer is not among the options provided.