Calculate the term containing x^3 in the expansion of (2x + 5)^6. (2000)

Calculate the term containing x^3 in the expansion of (2x + 5)^6. (2000)

Step 1: Identify the expression to expand, which is (2x + 5)^6.
Step 2: Recognize that we need to find the term that contains x^3.
Step 3: Use the binomial expansion formula, which is C(n, k) * (a)^k * (b)^(n-k), where n is the total number of terms, k is the power of x, a is the coefficient of x, and b is the other term.
Step 4: In our case, n = 6 (the exponent), k = 3 (the power of x we want), a = 2 (the coefficient of x), and b = 5.
Step 5: Calculate C(6, 3), which is the number of ways to choose 3 terms from 6. This equals 20.
Step 6: Calculate (2)^3, which is 2 * 2 * 2 = 8.
Step 7: Calculate (5)^(6-3), which is (5)^3 = 5 * 5 * 5 = 125.
Step 8: Multiply all the parts together: 20 * 8 * 125.
Step 9: Calculate 20 * 8 = 160.
Step 10: Then calculate 160 * 125 = 20000.
Step 11: The term containing x^3 in the expansion is 20000.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific terms in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the coefficients of the terms in the expansion.
Exponent Rules – Applying the rules of exponents to simplify the terms in the expansion.