Final answer:
The student is asking about the value of the expression combining three binomial coefficients: C(n,r+1), C(n,r-1), and C(2n,r). Without additional information, we cannot simplify this expression further as it involves combinations from different sets. The problem relates to the branch of mathematics known as combinatorics and utilizes the binomial theorem.
Step-by-step explanation:
The student's question involves working out the value of the expression C(n,r+1) + C(n,r-1) + C(2n,r). This type of problem falls under combinatorics, a branch of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
To solve the expression, we need to recall some properties of combinations and the binomial theorem. By Pascal's rule, which relates to the binomial coefficients, we have C(n, r) = C(n-1, r-1) + C(n-1, r). Using this, we can express C(n, r+1) as C(n, r+1) = C(n-1, r) + C(n-1, r+1) and C(n, r-1) as C(n, r-1) = C(n-1, r-2) + C(n-1, r-1). So when we add C(n, r+1) and C(n, r-1), the term C(n-1, r) appears in both expansions, allowing us to combine them. However, without additional context or restrictions on n and r, we can't simplify this expression to a more compact form. Usually, we expect some relationship between n and r that would let us simplify further.
For the term C(2n, r), we're looking at a different binomial coefficient entirely, which represents the number of ways to choose r items from a set of 2n items. Since it is not directly related to the other two terms without additional constraints or relationships, we can't combine it with them in a meaningful way.