I need some help I have to Select one problem from the self-practice exercises and teach the solution to your classmate as if they were an elementary school student. I have to explane it in full details.
The self-practice problem I chose was:
Suppose that the U.S. Senate consisted of 57 Republican and 43 Democratic Senators. How many different 12-person committees could be formed with 8 Republican senators and 4 democratic senators?
Sure! Let's break down the problem step by step.
To solve this problem, we need to use a concept called combinations. Combinations help us find out how many different ways we can choose a certain number of items from a larger group, without considering the order in which the items are selected.
In this case, we want to find out how many different 12-person committees can be formed with 8 Republican senators and 4 Democratic senators, given that there are 57 Republican senators and 43 Democratic senators in total.
Here's how we can solve it:
Step 1: First, we need to determine how many ways we can choose 8 Republican senators out of the 57 available Republican senators. We can do this by using the formula for combinations, which is written as "nCr". In our case, n is the total number of items (57 Republican senators) and r is the number of items to be chosen (8 Republican senators). So, we need to calculate 57C8.
Step 2: We can use the formula for combinations to calculate 57C8. The formula is nCr = n! / (r! * (n-r)!). Here, "!" denotes factorial, which means multiplying a number with all the positive integers less than itself.
For 57C8, we write it as 57! / (8! * (57-8)!). We can simplify the expression as 57! / (8! * 49!).
Step 3: To evaluate the factorial expressions, we multiply the numbers together. For example, to calculate 57!, we multiply 57 by all the positive integers less than 57. Similarly, we calculate 8! and 49!.
Step 4: Once we have computed the values for 57!, 8!, and 49!, we can substitute them back into the formula 57! / (8! * 49!).
Step 5: After evaluating this expression, we will get the number of different ways we can choose 8 Republican senators out of the total 57 Republican senators.
Step 6: Now, we repeat the same process to find out how many ways we can choose 4 Democratic senators out of the 43 available Democratic senators. We calculate 43C4 using the same formula and steps as before.
Step 7: Lastly, to find out the total number of different 12-person committees that can be formed with 8 Republican senators and 4 Democratic senators, we multiply the number of ways we can choose the Republican senators (from Step 5) by the number of ways we can choose the Democratic senators (from Step 6).
And voila! That gives us the final answer to the problem. By following these steps, we can determine the number of different 12-person committees that can be formed with 8 Republican senators and 4 Democratic senators.