Question

The number of ways of selecting five members to form a committe from 7 men and 10 women is (1) 5266 (2) 6123 (3) 6188 (4) 8123

Short answer

Answer: There are 6188 ways to select five members for the committee from a group of 7 men and 10 women.

Step-by-step solution

Combination 1: All five members are women

We need to choose 5 women from a group of 10 women: C(10, 5) = 10! / (5!5!) = 252

Combination 2: Four women, one man

We need to choose 4 women from 10, and 1 man from 7: C(10, 4) * C(7, 1) = (10! / (4!6!)) * (7! / (1!6!)) = 210 * 7 = 1470

Combination 3: Three women, two men

We need to choose 3 women from 10, and 2 men from 7: C(10, 3) * C(7, 2) = (10! / (3!7!)) * (7! / (2!5!)) = 120 * 21 = 2520

Combination 4: Two women, three men

We need to choose 2 women from 10, and 3 men from 7: C(10, 2) * C(7, 3) = (10! / (2!8!)) * (7! / (3!4!)) = 45 * 35 = 1575

Combination 5: One woman, four men

We need to choose 1 woman from 10, and 4 men from 7: C(10, 1) * C(7, 4) = (10! / (1!9!)) * (7! / (4!3!)) = 10 * 35 = 350

Combination 6: All five members are men

We need to choose 5 men from a group of 7 men: C(7, 5) = 7! / (5!2!) = 21 Now, we will add up all the combinations to find the total number of ways to select five members for the committee.

Calculate Total Number of Ways

Total number of ways = 252 + 1470 + 2520 + 1575 + 350 + 21 = 6188 So, the total number of ways to select five members from a group of 7 men and 10 women is 6188. The correct answer is (3) 6188.

Key concepts

Combination Formula

The combination formula is a fundamental tool in combinatorics, which helps us find the number of ways to choose a subset of items from a larger set. Unlike permutations, combinations do not consider the sequence or order of selection. This makes them perfect for problems where order doesn't matter, like selecting members for a committee.

The formula for combinations is often written as \( C(n, k) \), where \( n \) denotes the total number of items, and \( k \) represents the number of items to be chosen. It is given by:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

In our exercise, we used the combination formula multiple times to choose different configurations of men and women for the committee. For example, to select 5 members from 10 women, we calculated \( C(10, 5) \), and this gave us the number of possible selections without regard to order.

Permutation

Permutations are closely related to combinations but differ in that they consider the order of items in a set. When you need to arrange or sequence a group of items, permutations become essential. However, in problems like committee selection, permutations are not used since the order of selection does not matter.

The key contrast between combinations and permutations is that permutations focus on "arrangements," while combinations deal with "selections." For permutations, our formula looks slightly different:

• \( P(n, k) = \frac{n!}{(n-k)!} \)

Here, the sequence matters, so you'll end up with a much larger number of possibilities compared to combinations if order was important. It's important to use permutations in appropriate contexts such as seating arrangements or organizing events where order matters.

Committee Selection

Committee selection is a typical problem in combinatorics where combinations are applied. The goal is to choose a subset of members from a larger group. Key in these problems is recognizing that the order doesn't matter, which is why we use the combination formula instead of a permutation.

In our exercise, we explored different scenarios for committee selection:

• All five members are women
• Four women and one man
• Three women and two men
• Two women and three men
• One woman and four men
• All five members are men

Each scenario required calculating a number of combinations to find the possible ways to select committee members. By summing these up, we found the total number of ways to form the committee given the constraints.

Discrete Mathematics

Discrete mathematics is a vast field of mathematics dealing with countable, distinct, and separate collections of objects. Unlike continuous math, which deals with objects that can vary smoothly, discrete math revolves around separate and distinct values.

Combinatorics, including the topics of permutations and combinations, plays a fundamental role in discrete mathematics. It deals with counting, arrangement, and selection of objects in sets, which is crucial in solving problems like forming committees, planning events, and organizing resources efficiently.

Understanding discrete mathematics and its tools, such as the combination and permutation formulas, allows us to tackle real-world problems where precise counting and non-continuous structures are involved. It's widely applicable in computer science, cryptography, logistics, and other fields where distinct items are important.