find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls if 3 of the boys are cousins are either on the team or all not in the team.
Case 1: all the cousins are on the team ---> leaves 2 more to choose from the remaining 11 students = C(11,2) = 55
Case 2: none of the cousins are on the team --- leaves 5 to choose from the remaining 11 students = C(11,5) = 462
add them up
Well, well, well, it seems we have quite the comedic situation here! Let's take a look at how we can form this school team.
First, we have to consider the three cousin boys. Either all three are in the team or all three are out of the team. We can count each of these scenarios separately, and then add them up.
Scenario 1: All three cousin boys are in the team
If all three boys are in the team, we need to choose 2 more pupils from the remaining boys and girls. We have 3 boys (the cousins) and 5 girls left. So, we have to choose 2 pupils from a pool of 3 boys and 5 girls. This can be calculated as C(3, 2) * C(5, 2) = 3 * 10 = 30 ways.
Scenario 2: All three cousin boys are NOT in the team
If all three boys are not in the team, we need to choose 5 pupils from the remaining boys and girls. We have 3 boys (the cousins) and 8 girls left. So, we have to choose 5 pupils from a pool of 3 boys and 8 girls. This can be calculated as C(3, 0) * C(8, 5) = 1 * 56 = 56 ways.
Adding up the two scenarios, we get a total of 30 + 56 = 86 ways.
So, my friend, there are 86 ways of choosing a school team of 5 pupils from 6 boys and 8 girls, while making sure that either all 3 cousin boys are on the team or all 3 are not on the team!
Just remember, it's all fun and games until someone has to crunch some numbers!
To find the number of ways of choosing a school team of 5 pupils, we need to consider the cases where all 3 cousins are on the team and the cases where none of them are on the team.
Case 1: All 3 cousins are on the team.
In this case, we have to choose 2 more pupils from the remaining 6 boys (excluding the 3 cousins) and 8 girls.
The number of ways to choose 2 pupils from 6 boys is given by the combination formula: C(6, 2) = 6! / (2! * (6-2)!) = 15.
The number of ways to choose 2 pupils from 8 girls is given by the combination formula: C(8, 2) = 8! / (2! * (8-2)!) = 28.
Therefore, the number of ways to choose a team with all 3 cousins is 15 * 28 = 420.
Case 2: None of the 3 cousins are on the team.
In this case, we have to choose all 5 pupils from the remaining 3 boys and 8 girls.
The number of ways to choose 5 pupils from 3 boys is given by the combination formula: C(3, 5) = 3! / (5! * (3-5)!) = 0 (as we cannot choose 5 pupils from only 3 boys).
The number of ways to choose 5 pupils from 8 girls is given by the combination formula: C(8, 5) = 8! / (5! * (8-5)!) = 56.
Therefore, the number of ways to choose a team with none of the cousins is 0 * 56 = 0.
In conclusion, the total number of ways of choosing a team is 420 + 0 = 420 ways.
To find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls, we can use the concept of combinations.
First, let's consider the scenario where all 3 cousin boys are on the team. In this case, we need to choose 2 more boys from the remaining 3 boys who are not cousins. The number of ways to choose 2 boys from 3 non-cousin boys can be calculated using combinations:
C(n, k) = n! / (k!(n-k)!)
Where n is the total number of items, and k is the number of items to be chosen.
In this case, n = 3 (non-cousin boys) and k = 2, so we have:
C(3, 2) = 3! / (2!(3-2)!) = 3
Now, let's consider the scenario where the 3 cousin boys are not on the team. In this case, we need to choose 5 pupils from the remaining 3 boys and 8 girls. Again, we can use combinations to find the number of ways:
C(n, k) = n! / (k!(n-k)!)
Here, n = (3 non-cousin boys) + (8 girls) = 11, and k = 5, so we have:
C(11, 5) = 11! / (5!(11-5)!) = 11! / (5!6!) = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1) = 462
Finally, to determine the total number of ways to choose the school team, we need to add up the number of ways from both scenarios.
Total ways = (number of ways with the 3 cousin boys on the team) + (number of ways with the 3 cousin boys not on the team)
Total ways = 3 + 462 = 465
Therefore, there are 465 different ways of choosing a school team of 5 pupils from 6 boys and 8 girls, considering the given conditions.