How many commitees can be formed of 5 people from 20 men &  12 women if there has to be more men than the women on the committee 

To solve this problem, we need to calculate the number of committees of 5 people that can be formed such that the number of men is greater than the number of women. 

Given:
- 20 men
- 12 women

Let's consider all possible cases where the number of men is greater than the number of women:

1. **4 men and 1 woman**  
2. **3 men and 2 women**  
3. **5 men and 0 women**  

Now, we calculate the number of ways to form each type of committee:

### Case 1: 4 Men and 1 Woman
- Number of ways to choose 4 men from 20: \(\binom{20}{4}\)
- Number of ways to choose 1 woman from 12: \(\binom{12}{1}\)

Total for this case:  
\[
\binom{20}{4} \times \binom{12}{1} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} \times 12 = 4845 \times 12 = 58140
\]

### Case 2: 3 Men and 2 Women
- Number of ways to choose 3 men from 20: \(\binom{20}{3}\)
- Number of ways to choose 2 women from 12: \(\binom{12}{2}\)

Total for this case:  
\[
\binom{20}{3} \times \binom{12}{2} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} \times \frac{12 \times 11}{2 \times 1} = 1140 \times 66 = 75240
\]

### Case 3: 5 Men and 0 Women
- Number of ways to choose 5 men from 20: \(\binom{20}{5}\)
- Number of ways to choose 0 women from 12: \(\binom{12}{0} = 1\)

Total for this case:  
\[
\binom{20}{5} \times \binom{12}{0} = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} = 15504
\]

### Adding up all cases:
\[
58140 + 75240 + 15504 = 148884
\]

Thus, the total number of committees that can be formed with more men than women is **148,884**.

Would you like more details or have any questions? Here are some related questions:

1. How many ways can a committee of 5 be formed with at least 3 men?
2. What if the committee must have exactly 3 men and 2 women?
3. How many committees can be formed with exactly 2 men and 3 women?
4. What is the probability of randomly selecting a committee with more men than women?
5. How does the number of committees change if there are 25 men instead of 20?

**Tip:** In combinatorial problems, carefully consider the conditions given (e.g., "more men than women") to ensure you cover all valid cases.

Explore how to count the number of committees with more men than women formed from a group of 20 men and 12 women. Learn the application of combinatorial methods such as binomial coefficients. Detailed step-by-step solution provided.

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