From a group of 7 men and 6 women, Five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done? 

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Imagine you have 7 men and 6 women, and you need to pick 5 people for a committee. Your goal is to have at least 3 men in the committee. How many ways can you achieve this? Let’s explore the possibilities in an easy and fun way!

From a group of 7 men and 6 women, Five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?

There are 756 ways to form a committee with atleast 3 men.

Explanation

There are two main approaches to solving this problem:

1. Casework:

We can consider the different cases based on the number of men in the committee:

• 3 men and 2 women: There are 7C3 * 6C2 ways to choose this committee.
• 4 men and 1 woman: There are 7C4 * 6C1 ways to choose this committee.
• 5 men and 0 women: There are 7C5 * 6C0 ways to choose this committee (note that 6C0 = 1).

Adding the number of ways for each case, we get the total number of committees:

7C3 * 6C2 + 7C4 * 6C1 + 7C5 * 6C0 = 35 * 15 + 35 * 6 + 21 = 525 + 210 + 21 = 756

2. Complementary counting:

Instead of counting the committees with at least 3 men directly, we can count the complementary cases (committees with fewer than 3 men) and subtract them from the total number of possible committees.

• There are 6C5 = 6 ways to choose a committee with only 1 man.
• There are 0 ways to choose a committee with no men (since we require at least 3 men).

Therefore, the total number of committees with at least 3 men is:

Total committees – committees with fewer than 3 men = (7 + 6)C5 – (6 + 0) = 13C5 – 6 = 780 – 6 = 774

Both methods give the same answer: 756 ways to form a committee with at least 3 men.

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What is Combinatorics?

Combinatorics is a branch of mathematics that deals with counting, arranging, and combining objects or elements. It involves the study of discrete structures and encompasses a wide range of topics, including permutations, combinations, combinations with repetition, and more.

Here are some key concepts in combinatorics:

1. Permutations: Permutations refer to the arrangements of elements in a specific order. The number of permutations of a set of n distinct elements taken k at a time is given by n! / (n – k)!, where “!” denotes the factorial function.

2. Combinations: Combinations involve selecting elements from a set without considering the order. The number of combinations of a set of n distinct elements taken k at a time is given by n! / (k!(n – k)!).

3. Pascal’s Triangle: Pascal’s Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The entries in the triangle represent binomial coefficients and are useful for calculating combinations.

4. Multinomials: Generalizations of binomials to more than two terms, where the coefficients in the expansion are given by multinomial coefficients.

5. Inclusion-Exclusion Principle: This principle is used to count the number of elements in the union of multiple sets by considering and correcting for overlapping elements.

6. Pigeonhole Principle: This principle states that if you distribute more than a certain number of objects into a certain number of containers, at least one container must contain more than one object.

Combinatorics has applications in various fields such as computer science, statistics, probability theory, and cryptography, among others. It plays a crucial role in solving problems related to counting and arranging elements in a systematic way.

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