find the coefficient of the fourth term of (×+2)^5

Question

a. 32
b. 48
c. 64
d. 80

Answer 1

The coefficient of the fourth term of (x+2)^5 can be found using the binomial theorem.

The binomial theorem states that the coefficient of the term with x^k in the expansion of (a+b)^n is given by:

C(n, k) * a^(n-k) * b^k

Where C(n, k) is the binomial coefficient, which can be found using the formula:

C(n, k) = n! / (k! * (n-k)!)

In this case, we want to find the coefficient of the fourth term, so k = 4. The exponent of x is (n-k) = (5-4) = 1.

Using the binomial coefficient formula:

C(5, 4) = 5! / (4! * (5-4)!) = 5

Now, substituting n = 5, k = 4, a = x, and b = 2 into the binomial theorem formula:

C(5, 4) * x^(5-4) * 2^4 = 5 * x^1 * 2^4 = 10x

Therefore, the coefficient of the fourth term of (x+2)^5 is 10.

None of the given answer choices (a, b, c, or d) matches 10.