Final answer:
In the binomial expansion of (x + y)^10, the terms are of the form ax^m*y^(10-m). The options ax^6y^4 and ax^4y^6 are present in the expansion because their exponents sum to 10, while ax^5 and ax^3y^7 are not because they do not meet this condition.
Step-by-step explanation:
The question involves the expansion of a binomial expression (x + y)^10, which follows the binomial theorem. To find the terms that appear in the expansion, we need to apply the theorem to each term in the sequence. The coefficients of the expansion follow the pattern of Pascal's triangle, which gives the coefficients for the binomial terms in the expansion of (x + y)^n where n is a positive integer.
In the binomial expansion of (x + y)^10, the general term is given by Tk+1 = C(n, k) * xn-k * yk, where C(n, k) is the binomial coefficient, which is the k-th element in the n-th row of Pascal's triangle. When n = 10, the possible terms will include x raised to a power plus y raised to another power with the condition that the sum of the exponents equals 10. Thus, the terms present in the expansion will be of the form axmy10-m where m is an integer ranging from 0 to 10.
Based on the options given:
- A. ax^6y^4 is correct because 6 + 4 = 10.
- B. ax^5 is incorrect because it does not have a corresponding y term such that the sum of the exponents equals 10.
- C. ax^4y^6 is correct because 4 + 6 = 10.
- D. ax^3y^7 is incorrect because it is not a term in the expansion; there is no ax^3 term in the expansion of (x + y)^10 as the corresponding coefficient would be zero.