### Investigative Team SelectionThe state narcotics bureau must form a 5-member investigative team. If it has 30 agents from which to choose, how many different possible teams can be formed?**Question:** There are ____ ways to choose a team of 5 out of 30 agents.**Explanation for Website:**In this scenario, we need to determine how many distinct 5-member teams can be formed from a pool of 30 agents. This is a classic combination problem in combinatorics, where the order of selection does not matter.To find the number of combinations, we use the combination formula:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]where:- $ n $ is the total number of items (agents, in this case),- $ k $ is the number of items to choose (team members).Plugging in the values:\[ C(30, 5) = \frac{30!}{5!(30-5)!} \]Calculating this will give the number of different ways to choose a team of 5 out of 30 agents.### Graphs and Diagrams*Note:* There are no diagrams or graphs in this image. The image merely has a blank space for the answer and two buttons, one with an 'X' and one with a circular arrow indicating reset, likely for clearing and trying again.

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Investigative Team The state narcotics bureau must form a 5-member investigative team. If it has 30 agents from which to choose, how many different possible teams can be formed? There are ways to choose a team of 5 out of 30 agents.

Investigative Team The state narcotics bureau must form a 5-member investigative team. If it has 30 agents from which to choose, how many different possible teams can be formed? There are ways to choose a team of 5 out of 30 agents.

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Concept explainers

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.

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### Investigative Team Selection

The state narcotics bureau must form a 5-member investigative team. If it has 30 agents from which to choose, how many different possible teams can be formed?

**Question:**
There are ____ ways to choose a team of 5 out of 30 agents.

**Explanation for Website:**

In this scenario, we need to determine how many distinct 5-member teams can be formed from a pool of 30 agents. This is a classic combination problem in combinatorics, where the order of selection does not matter.

To find the number of combinations, we use the combination formula:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

where:
- $ n $ is the total number of items (agents, in this case),
- $ k $ is the number of items to choose (team members).

Plugging in the values:

\[ C(30, 5) = \frac{30!}{5!(30-5)!} \]

Calculating this will give the number of different ways to choose a team of 5 out of 30 agents.

### Graphs and Diagrams

*Note:* There are no diagrams or graphs in this image. The image merely has a blank space for the answer and two buttons, one with an 'X' and one with a circular arrow indicating reset, likely for clearing and trying again.

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