How many different teams of 3 can be chosen from a group of 14 adults and 12 children if each team must have at least two children on it?
12C2 ways to pick the first two kids
There are 14+10 = 24 ways to pick the 3rd person
12C2 * 24
Well, if we're creating teams with at least two children, we can start by selecting two children and one adult for each team. Let's do the math: there are 12 children to choose from, so we have (12 choose 2) = 66 ways to pick two children.
And for each pair of children, we have 14 adults to choose from, so there are 14 possibilities. Multiply these together (66 x 14) and we get 924 different teams that can be chosen. Quite a bunch of mini teammates!
To calculate the number of different teams of 3 that can be chosen from a group of 14 adults and 12 children, where each team must have at least two children, we can break down the problem into two cases:
Case 1: One adult and two children in a team
In this case, we need to select one adult from a group of 14 adults, and two children from a group of 12 children. Since the order of selection does not matter, we need to calculate the number of combinations.
Selecting one adult from 14: C(14, 1) = 14
Selecting two children from 12: C(12, 2) = 66
To calculate the total number of teams in this case, we multiply the number of options for selecting one adult and two children.
Total number of teams in Case 1: 14 * 66 = 924
Case 2: Two adults and one child in a team
In this case, we need to select two adults from a group of 14 adults, and one child from a group of 12 children.
Selecting two adults from 14: C(14, 2) = 91
Selecting one child from 12: C(12, 1) = 12
To calculate the total number of teams in this case, we multiply the number of options for selecting two adults and one child.
Total number of teams in Case 2: 91 * 12 = 1092
To find the total number of teams that can be chosen, we sum up the number of teams from both cases.
Total number of teams: 924 + 1092 = 2016
Therefore, there are 2016 different teams of 3 that can be chosen from a group of 14 adults and 12 children, where each team must have at least two children.
To find the number of different teams of 3 that can be chosen from a group of 14 adults and 12 children, where each team must have at least two children on it, we can use the concept of combinations.
Step 1: Count the number of ways to choose 2 children and 1 adult.
- This can be calculated using combinations: C(12, 2) * C(14, 1) = (12! / (2! * 10!)) * (14! / (1! * 13!))
- C(n, r) represents the number of combinations of choosing r items from a group of n items.
- In this case, we choose 2 children (12! / (2! * 10!)) and 1 adult (14! / (1! * 13!)).
Step 2: Count the number of ways to choose 3 children only.
- This can be calculated using combinations: C(12, 3) = 12! / (3! * 9!).
Step 3: Add the results from Step 1 and Step 2 to find the total number of different teams.
- Total number of different teams = Step 1 + Step 2
Therefore, the number of different teams of 3 that can be chosen is the sum of C(12, 2) * C(14, 1) and C(12, 3).