Final answer:
The coefficient of the fourth term in the expansion of (x+5)^7 is obtained using the binomial theorem, which gives us 35 * 125 = 4375. However, since that number is not an option given, it seems there's a mistake. The closest given answer is 35, the binomial coefficient without considering the factor of 5 cubed.
Step-by-step explanation:
To find the coefficient of the fourth term in the expansion of (x+5)^7, we need to use the binomial theorem. The binomial theorem allows us to expand expressions of the form (a + b)^n, where 'n' is a non-negative integer. In such an expansion, the general term is given by the formula:
T(k+1) = C(n, k) * a(n-k) * b^k,
where C(n, k) is the binomial coefficient n! / (k! * (n-k)!), often read as 'n choose k'.
The fourth term corresponds to k = 3 (since we start counting from k = 0), so the formula for the coefficient becomes:
C(7, 3) * x7-3) * 53 = C(7, 3) * x4 * 125.
The binomial coefficient C(7, 3) is equal to 7! / (3! * 4!) = 35. Therefore, the coefficient of the fourth term is:
35 * 125 = 4375.
However, this is not an option in the choices provided by the student, indicating a possible mistake in the question or options (since 35 is the binomial coefficient before multiplying by 53). The closest answer to the provided options, without the full calculation, is Option A (35), which is the binomial coefficient without considering the multiplicative factor of 5 cubed.