Binomial Coefficients in the Expansion of (x + a)^n

In the expansion of (x + a)^n, the binomial coefficients represent the coefficients of each term. The binomial coefficient for the term with x^k will be denoted as C(n, k), where n is a positive integer and k is a non-negative integer less than or equal to n.

Equidistant Binomial Coefficients

The term "equidistant from the beginning and the end" refers to the binomial coefficients that are symmetrically located in the expansion. In other words, these coefficients have the same distance from the first and last terms.

For example, in the expansion of (x + a)^5, the binomial coefficients C(5, 1) and C(5, 4) are equidistant from the beginning and the end. Similarly, C(5, 2) and C(5, 3) are also equidistant.

Additive Inverse Property

The additive inverse property states that for any real number x, there exists another real number -x such that their sum is zero. In other words, x + (-x) = 0.

In the case of equidistant binomial coefficients, if C(n, k) and C(n, n-k) are equidistant from the beginning and the end, then their sum will be zero. This can be observed by considering the Pascal's triangle, which shows that the binomial coefficients on the same diagonal are equal but have opposite signs.

Multiplicative Inverse Property

The multiplicative inverse property states that for any non-zero real number x, there exists another real number 1/x such that their product is one. In other words, x * (1/x) = 1.

In the case of equidistant binomial coefficients, if C(n, k) and C(n, n-k) are equidistant from the beginning and the end, then their product will be one. This can be observed by considering the combinatorial interpretation of binomial coefficients, which represents the number of ways to choose k objects from a set of n objects.

Conclusion

In the expansion of (x + a)^n, the equidistant binomial coefficients are additive inverses of each other. This means that their sum is zero. They are not multiplicative inverses of each other, as their product is not necessarily one. Therefore, the correct answer is option 'C': equidistant binomial coefficients are equal.