Final answer:
A committee of two democrats and three republicans can be formed in 4620 different ways from a group of seven democrats and twelve republicans, using combinations to calculate the possibilities.
Step-by-step explanation:
To determine in how many ways a committee of two democrats and three republicans can be formed from a group of seven democrats and twelve republicans, we use combinations because the order in which we select the committee members does not matter.
We calculate the number of ways to choose two democrats out of seven and multiply this by the number of ways to choose three republicans out of twelve.
The number of ways to choose two democrats out of seven is calculated using the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of items to pick from, k is the number of items to pick, and ! denotes factorial. Applying this formula, we have C(7, 2) for democrats which equals 7! / (2! * 5!) = 21 ways.
Similarly, the number of ways to choose three republicans out of twelve is C(12, 3) which equals 12! / (3! * 9!) = 220 ways.
Finally, to find the total number of ways to form the committee, we multiply the two results: 21 ways * 220 ways = 4620 ways.