In how many ways can a committee of five Democrats and four Republicans be formed from a group of twelve Democrats and eleven Republicans?
The number of ways to form a committee of 5 Democrats from a group of 12 Democrats is given by the combination formula:
${{12}\choose{5}} = \frac{12!}{5!(12-5)!} = 792$
Similarly, the number of ways to form a committee of 4 Republicans from a group of 11 Republicans is:
${{11}\choose{4}} = \frac{11!}{4!(11-4)!} = 330$
Therefore, the total number of ways to form a committee of 5 Democrats and 4 Republicans is:
$792 \times 330 = 261,360$
So, there are 261,360 ways to form a committee of five Democrats and four Republicans from a group of twelve Democrats and eleven Republicans.