Question

Thirty members attended a party. If each person shakes hands with every other person exactly once, then find the number of hand shakes made in the party. (1) \({ }^{30} \mathrm{P}_{2}\) (2) \({ }^{30} \mathrm{C}_{2}\) (3) \({ }^{29} \mathrm{C}_{2}\) (4) \({ }^{60} \mathrm{C}_{2}\)

Short answer

Answer: 435 handshakes

Step-by-step solution

Understanding the concept of combinations

In a combination, we find the number of ways to select a certain number of objects from a larger set, without regard to the order of the selection. In this case, we need to find the total number of ways people can make handshakes with one another without concerning the order of the handshakes.

Identify the correct formula

We will use the formula for combinations: \[ ^nC_k = \frac{n!}{k!(n-k)!} \] where n is the total number of objects, k is the number of objects to be selected, and ! denotes the factorial.

Apply the formula to the given problem

In this problem, we have 30 people (n = 30) and we need to select 2 people at a time for a handshake (k = 2). Applying the combination formula to solve the problem: \[ ^{30}C_2 = \frac{30!}{2!(30-2)!} \]

Simplify and calculate the result

Now, let's simplify and calculate the value: \[ ^{30}C_2 = \frac{30!}{2!(28)!} = \frac{30 \times 29}{2 \times 1} = 15 \times 29 = 435 \] So, the answer is option (2) \({ }^{30} \mathrm{C}_{2}\) and the number of handshakes made in the party is 435.

Key concepts

Factorial Notation

Understanding factorial notation is crucial when dealing with permutations, combinations, and various problems in combinatorics. A factorial, represented by an exclamation mark (!), refers to the product of all positive integers up to a certain number.

For example, the factorial of 5, written as 5!, is calculated as:
\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

When using factorial notation in combinatorial problems, it's helpful to remember that the factorial of zero (0!) is defined as 1, and that factorials grow very rapidly with larger numbers. Factorial notation is foundational in calculating the number of ways objects can be arranged (permutations) or selected (combinations), and a clear understanding can simplify the process of solving these problems considerably.

Permutations and Combinations

Permutations and combinations are two concepts in mathematics that deal with the arrangement and selection of objects without repetition. While they seem similar, there is a fundamental difference between the two.

Permutations focus on arranging objects in a specific order. The order is important and changing the order creates a new permutation. For instance, the different ways to arrange the letters A, B, and C are ABC, ACB, BAC, BCA, CAB, and CBA.

Combinations, on the other hand, are about selecting objects where the order does not matter.

Using the same letters A, B, and C, the combinations of choosing 2 letters would be AB, BC, and AC. Notice that BA is not listed because it is considered the same as AB in combinations.

The formula for combinations identifies the number of ways to select a subset of objects from a set without regard to their order. When we apply combinations to the context of our party handshake problem, we are simply counting each unique pair of people who shake hands, regardless of the sequence.

Combinatorial Problem Solving

Combinatorial problem solving involves using strategies and mathematical concepts like permutations and combinations to solve problems that require counting the number of ways to arrange or select items from a set.

In problems like the party handshake scenario, where each guest shakes hands with every other guest exactly once, we have a classic example of a combinational problem. Here, we're not concerned with the order of handshakes, only the fact that they occur. To solve this, we engage combinatorial reasoning to recognize that each handshake is a combination of two people.

Effective problem solving often depends on identifying the correct approach.

In this case, we use combinations rather than permutations because the order in which two people shake hands is irrelevant; ‘Person A to Person B’ is the same as ‘Person B to Person A’. This understanding simplifies the problem greatly and leads us directly to the correct mathematical formula to use, which is the combinations formula.

By mastering combinatorial problem-solving techniques, you will be well-equipped to tackle a wide range of mathematical challenges, from simple to complex.