In how many ways can a committee consisting of 4 Pharmacists and 3 Biologists be selected from 9 Pharmacists and 8 Biologists, if: There are no restrictions. A particular Pharmacis... In how many ways can a committee consisting of 4 Pharmacists and 3 Biologists be selected from 9 Pharmacists and 8 Biologists, if: There are no restrictions. A particular Pharmacist must be selected. A particular Biologist cannot serve on the committee. A particular Pharmacist and Biologist must serve on the committee.
Understand the Problem
The question is asking for the number of ways to form a committee under various conditions. It involves combinatorial selections from groups of Pharmacists and Biologists.
Answer
The total number of ways to form the committee is given by $$ \text{Total} = C(P, 2) \cdot C(B, 1) + C(P, 1) \cdot C(B, 2) $$
Answer for screen readers
The total number of ways to form the committee is given by:
$$ \text{Total} = C(P, 2) \cdot C(B, 1) + C(P, 1) \cdot C(B, 2) $$
Steps to Solve
- Identify the Groups and Requirements
We have two groups: Pharmacists and Biologists. Let's define:
- Number of Pharmacists = $P$
- Number of Biologists = $B$
- Determine Combinations for Each Scenario
We need to consider different scenarios for forming the committee.
- Calculating Combinations
For a hand-picked selection, we can use combinations represented by the binomial coefficient:
The number of ways to choose $k$ individuals from $n$ individuals is given by:
$$ C(n, k) = \frac{n!}{k!(n-k)!} $$
- Scenario Analysis
Let's analyze specific scenarios based on the prompt:
- Scenario 1: Selecting 2 Pharmacists and 1 Biologist
- Scenario 2: Selecting 1 Pharmacist and 2 Biologists
For Scenario 1:
$$ C(P, 2) \cdot C(B, 1) $$
For Scenario 2:
$$ C(P, 1) \cdot C(B, 2) $$
- Total Combinations
Finally, we sum the combinations from both scenarios:
$$ \text{Total} = C(P, 2) \cdot C(B, 1) + C(P, 1) \cdot C(B, 2) $$
The total number of ways to form the committee is given by:
$$ \text{Total} = C(P, 2) \cdot C(B, 1) + C(P, 1) \cdot C(B, 2) $$
More Information
Combinatorial problems often involve the use of binomial coefficients to determine the number of ways to choose groups from larger sets. In real-world applications, this can represent how teams or committees are formed in professional settings.
Tips
- Forgetting About the Factorial: Not using $!$ when calculating combinations can lead to incorrect answers. Always remember to compute factorials correctly.
- Overlapping Selections: When considering multiple scenarios, make sure that selections do not overlap unless specified in the problem.
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