A debate team of 4 is to be chosen from a class of 29 where the order in which the members of the team are chosen does not matter. There are two twin brothers in the class. How many possible ways can the team be formed which will not include any of the twin brothers?
excluding the twins means only 27 people are available.
So, C(27,4) is the number of teams possible.
To find the number of possible ways to form a debate team of 4 from a class of 29 where the order does not matter, we need to use the concept of combinations.
First, let's determine the number of ways to choose 4 students from a class of 29 without any restrictions. This can be calculated using the formula for combinations:
nCr = n! / (r! * (n-r)!)
Where n is the total number of students in the class (29 in this case) and r is the number of students to be chosen (4 in this case), and "!" denotes the factorial function.
Using this formula, we can calculate the number of ways to form a team without any restrictions:
29C4 = 29! / (4! * (29-4)!)
Simplifying this:
29C4 = 29! / (4! * 25!)
Next, let's consider the restriction that the team should not include the twin brothers. Since the twin brothers are two individuals, we need to subtract the number of teams that include them from the total number of teams.
To calculate the number of teams that include the twin brothers, we need to choose 2 more students from the remaining 27 students (excluding the twin brothers).
27C2 = 27! / (2! * (27-2)!)
Simplifying this:
27C2 = 27! / (2! * 25!)
Now, let's subtract the number of teams that include the twin brothers from the total number of teams, which we found earlier:
Total number of teams excluding twin brothers = 29C4 - 27C2
Therefore, the number of possible ways to form a team of 4 without including any of the twin brothers is:
29C4 - 27C2 = (29! / (4! * 25!)) - (27! / (2! * 25!))
Now, substitute the factorial values into the equation and simplify to get the final answer.
To determine the number of possible ways to form a debate team of 4 that does not include any of the twin brothers, we need to subtract the number of ways to choose the team with at least one or both of the twin brothers from the total number of possibilities.
Step 1: Find the total number of ways to choose a team of 4 from 29 students, without any restrictions on the twin brothers being chosen.
This can be calculated using the combination formula, denoted as C(n, r):
C(29, 4) = 29! / (4! * (29-4)!) = 29! / (4! * 25!)
= (29 * 28 * 27 * 26) / (4 * 3 * 2 * 1)
= 20475
Step 2: Find the number of ways to form a team that includes at least one or both of the twin brothers.
To calculate this, we'll subtract the number of ways to form a team with only one twin brother chosen from the total possibilities, and then subtract the number of ways to form a team with both twin brothers chosen.
- Number of ways to form a team with only one twin brother chosen:
There are 2 twin brothers, and we need to choose 1 out of the 29 students as a teammate, excluding the other twin brother. Therefore, this can be calculated as:
2 * C(28, 3) = 2 * (28! / (3! * (28-3)!)) = 2 * 3276 = 6552
- Number of ways to form a team with both twin brothers chosen:
Since the twin brothers are considered together, we only need to choose 2 more students from the remaining 27 students. Therefore, this can be calculated as:
C(27, 2) = 27! / (2! * (27-2)!) = (27 * 26) / (2 * 1) = 351
Step 3: Subtract the calculated values from the total possibilities.
Total possibilities without any restrictions: 20475
Ways to form a team with at least one twin brother chosen: 6552 + 351 = 6903
Number of possible ways to form a team of 4 without any twin brothers:
20475 - 6903 = 13572
Therefore, there are 13,572 possible ways to form a team of 4 without including any of the twin brothers.