To solve the problem step by step, we will analyze the given functions and expressions systematically.
### Step 1: Understanding the Binomial Coefficient
The binomial coefficient is defined as:
\[
{(s \choose r)} = \frac{s!}{r!(s-r)!} \quad \text{if } r \leq s \text{ and } r \geq 0
\]
This represents the number of ways to choose \( r \) elements from a set of \( s \) elements.
**Hint:** Recall the properties of binomial coefficients, especially how they relate to combinations.
### Step 2: Define the Function \( g(m, n) \)
We have:
\[
g(m, n) = \sum_{p=0}^{m+n} \frac{f(m, n, p)}{(n+p \choose p)}
\]
where \( f(m, n, p) \) is defined as:
\[
f(m, n, p) = \sum_{i=0}^{p} {(m \choose i)(n+i \choose i)(p+n \choose p-i)}
\]
**Hint:** Break down the summation into manageable parts, focusing on the inner function \( f(m, n, p) \).
### Step 3: Expand \( f(m, n, p) \)
We can rewrite \( f(m, n, p) \) as:
\[
f(m, n, p) = \sum_{i=0}^{p} mCi \cdot (n+i)Ci \cdot (p+n)C(p-i)
\]
This involves three binomial coefficients, which can be interpreted combinatorially.
**Hint:** Consider the combinatorial interpretation of each term in the summation.
### Step 4: Simplifying \( g(m, n) \)
Substituting \( f(m, n, p) \) into \( g(m, n) \):
\[
g(m, n) = \sum_{p=0}^{m+n} \frac{1}{(n+p \choose p)} \sum_{i=0}^{p} mCi \cdot (n+i)Ci \cdot (p+n)C(p-i)
\]
This can be simplified further by recognizing patterns in the summation.
**Hint:** Look for ways to combine or rearrange the summations to simplify the expression.
### Step 5: Recognizing Patterns
Notice that the summation over \( p \) and \( i \) can be interpreted as counting ways to distribute items. The final result can be expressed in terms of a known combinatorial identity.
**Hint:** Try to relate the result to known binomial expansions, such as \( (1+x)^{m+n} \).
### Step 6: Final Result
After simplifications, we find:
\[
g(m, n) = 2^{m+n}
\]
This means that \( g(m, n) \) counts the total number of subsets of a set with \( m+n \) elements.
**Hint:** Verify the result by checking specific values of \( m \) and \( n \).
### Step 7: Verifying Statements
Now we can check the statements provided in the question:
1. \( g(m, n) = 2^{m+n} \) is true.
2. \( g(m, n) = g(m, n+1) \) is also true.
3. \( g(2n) = 4^{n} \) is true.
**Hint:** Plug in values for \( m \) and \( n \) to validate each statement.
### Conclusion
The true statements are:
- \( g(m, n) = 2^{m+n} \)
- \( g(m, n) = g(m, n+1) \)
- \( g(2n) = 4^{n} \)
Thus, the answer is A, B, and D are true.
