In How many ways can a committee of 5 be formed from 5 juniors and 7 seniors If the committee must have 3 seniors?

There are 350 ways to form a committee of 5 with 3 seniors and 2 juniors.

To form a committee of 5 with 3 seniors and 2 juniors, we need to calculate the number of ways to choose 3 seniors out of 7 and 2 juniors out of 5, then multiply these combinations together.

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The number of ways to choose 3 seniors out of 7 is calculated using the combination formula:

7 choose 3, which equals 7! / (3! * (7-3)!) = (7 * 6 * 5) / (3 * 2 * 1) = 35.

Similarly, the number of ways to choose 2 juniors out of 5 is:

5 choose 2, which equals 5! / (2! * (5-2)!) = (5 * 4) / (2 * 1) = 10.

To find the total number of ways to form the committee, we multiply these two combinations:

Total ways = (7 choose 3) * (5 choose 2) = 35 * 10 = 350.

So, there are 350 ways to form a committee of 5 with 3 seniors and 2 juniors.

Combinations and Permutations

Combinations and permutations are fundamental concepts in combinatorics, a branch of mathematics dealing with counting, arrangement, and combination of objects. Here's a brief explanation of each:

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Permutations: Permutations refer to the arrangements of objects where the order matters. For example, consider the word "ABC." If you permute these letters, you could get "ABC," "ACB," "BAC," "BCA," "CAB," and "CBA." Each arrangement is considered a different permutation. The formula for permutations is given by:

P(n, r) = n! / (n - r)!

Where:

n is the total number of objects.
r is the number of objects taken at a time.
n! denotes factorial of n, which is the product of all positive integers up to n.

Combinations: Combinations, on the other hand, are selections of items where the order doesn't matter. Using the same example of the letters "ABC," the combinations would be "ABC," "ACB," and "BCA," but not "BAC," "CAB," or "CBA." The formula for combinations is given by:

C(n, r) = n! / (r! * (n - r)!)

Where:

n is the total number of objects.
r is the number of objects taken at a time.
n! denotes factorial of n, which is the product of all positive integers up to n.

In summary, permutations deal with the ordering of objects, while combinations focus on the selection of objects regardless of their order. Both concepts are crucial in various fields such as probability, statistics, and algorithm design.