Question

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is

Answer 1

Correct Answer: 5626

We need to select 4 men (M) and 4 women (W) from the two groups. Consider the following cases:

\(\text{From Group A}\)	\(\text{From Group B}\)	\(\text{Ways of Selection}\)
4M	4W	\({{4}\choose{4}} \cdot {{4}\choose{4}} = 1\)
3M1W	1M3W	\({{4}\choose{3}} \cdot {{5}\choose{1}} \cdot {{5}\choose{3}} \cdot {{4}\choose{1}} = 400\)
2M2W	2M2W	\({{4}\choose{2}} \cdot {{5}\choose{2}} \cdot {{5}\choose{2}} \cdot {{4}\choose{2}} = 3600\)
1M3W	3M1W	\({{4}\choose{1}} \cdot {{5}\choose{3}} \cdot {{5}\choose{1}} \cdot {{4}\choose{3}} = 1600\)
4W	4M	\({{5}\choose{4}} \cdot {{5}\choose{4}} = 25\)
Total		5626

Final Answer: 5626.

Answer 2

Group A: 4 men, 5 women
Group B: 5 men, 4 women
Choose 4 people from each group such that there are exactly 4 men and 4 women in total.
Step 1: Let the number of men selected from Group A be \( k \)
Then number of women from Group A = \( 4 - k \)
Number of men selected from Group B = \( 4 - k \)
Number of women from Group B = \( k \)
Since men in Group A = 4 and women in Group B = 4, \( k \) must lie between
\[ \max(0, 4 - 5) = 0 \quad \text{to} \quad \min(4, 4) = 4 \]
Step 2: Calculate the ways for each \( k \)
\[ \text{Ways} = \sum_{k=0}^4 \binom{4}{k} \binom{5}{4-k} \binom{5}{4 - k} \binom{4}{k} \] where
- \( \binom{4}{k} \) men from Group A
- \( \binom{5}{4-k} \) women from Group A
- \( \binom{5}{4-k} \) men from Group B
- \( \binom{4}{k} \) women from Group B
Calculate terms:
- For \( k=0 \): \( \binom{4}{0} \binom{5}{4} \binom{5}{4} \binom{4}{0} = 1 \times 5 \times 5 \times 1 = 25 \)
- For \( k=1 \): \( \binom{4}{1} \binom{5}{3} \binom{5}{3} \binom{4}{1} = 4 \times 10 \times 10 \times 4 = 1600 \)
- For \( k=2 \): \( \binom{4}{2} \binom{5}{2} \binom{5}{2} \binom{4}{2} = 6 \times 10 \times 10 \times 6 = 3600 \)
- For \( k=3 \): \( \binom{4}{3} \binom{5}{1} \binom{5}{1} \binom{4}{3} = 4 \times 5 \times 5 \times 4 = 400 \)
- For \( k=4 \): \( \binom{4}{4} \binom{5}{0} \binom{5}{0} \binom{4}{4} = 1 \times 1 \times 1 \times 1 = 1 \)
Sum: \[ 25 + 1600 + 3600 + 400 + 1 = 5626 \] Final answer: 5626