1. Understand the condition

We must select:

• $4$ persons from Group A (which has $4$ men, $5$ women),
• $4$ persons from Group B (which has $5$ men, $4$ women),

such that among the total $8$ selected persons, there are exactly:

• $4$ men and
• $4$ women.

So if we select $m_A$ men from Group A and $m_B$ men from Group B, then $m_A + m_B = 4.$ Also, since $4$ persons are chosen from each group,

• women chosen from Group A $= 4 - m_A$,
• women chosen from Group B $= 4 - m_B$.

2. List all possible distributions of men selected from the two groups

Since $m_A + m_B = 4$, the possible cases are: $(m_A,m_B)=(0,4),(1,3),(2,2),(3,1),(4,0).$

We count each case separately.

3. Case-wise counting

Case 1: $(m_A,m_B)=(0,4)$

• From Group A: choose $0$ men from $4$ and $4$ women from $5$.
• From Group B: choose $4$ men from $5$ and $0$ women from $4$.

Number of ways: $\binom{4}{0}\binom{5}{4}\binom{5}{4}\binom{4}{0}=1\cdot 5\cdot 5\cdot 1=25.$

Case 2: $(m_A,m_B)=(1,3)$

• From Group A: choose $1$ man from $4$ and $3$ women from $5$.
• From Group B: choose $3$ men from $5$ and $1$ woman from $4$.

Number of ways: $\binom{4}{1}\binom{5}{3}\binom{5}{3}\binom{4}{1}=4\cdot 10\cdot 10\cdot 4=1600.$

Case 3: $(m_A,m_B)=(2,2)$

• From Group A: choose $2$ men from $4$ and $2$ women from $5$.
• From Group B: choose $2$ men from $5$ and $2$ women from $4$.

Number of ways: $\binom{4}{2}\binom{5}{2}\binom{5}{2}\binom{4}{2}=6\cdot 10\cdot 10\cdot 6=3600.$

Case 4: $(m_A,m_B)=(3,1)$

• From Group A: choose $3$ men from $4$ and $1$ woman from $5$.
• From Group B: choose $1$ man from $5$ and $3$ women from $4$.

Number of ways: $\binom{4}{3}\binom{5}{1}\binom{5}{1}\binom{4}{3}=4\cdot 5\cdot 5\cdot 4=400.$

Case 5: $(m_A,m_B)=(4,0)$

• From Group A: choose $4$ men from $4$ and $0$ women from $5$.
• From Group B: choose $0$ men from $5$ and $4$ women from $4$.

Number of ways: $\binom{4}{4}\binom{5}{0}\binom{5}{0}\binom{4}{4}=1\cdot 1\cdot 1\cdot 1=1.$

4. Add all cases

Total number of ways: $25+1600+3600+400+1=5626.$

5. Final answer

The required number of ways is $\boxed{5626}.$

6. Comparison with stored correct answer

Stored correct answer = $5626$.

This matches our derived answer.