Six male and six female dancers perform the Virginia reel. This dance requires that they form a

step 1 Problem 31 fundamental counting principle if one event can occur in M ways and another event can

Six male and six female dancers perform the Virginia reel....

Key Concepts

Internal Arrangement of Pairs

In many pairing problems it is important to recognize that the order of individuals within each pair may matter, especially when specific roles or positions are significant. For each male-female pair, there are typically two possible orders, leading to a factor that is exponential in the number of pairs (2 raised to the number of pairs), which contributes to the overall count of arrangements.

Multiplication Principle

The multiplication principle in combinatorics states that if there are several independent choices or stages in an arrangement, the total number of outcomes is the product of the number of outcomes at each stage. In this scenario, the number of ways to pair the dancers, arrange the pairs in a line, and also consider the internal arrangement of each pair multiply together to give the total number of arrangements.

Permutations

Permutations refer to the different ways in which a set of objects can be arranged in order. In this problem, after the pairs are formed, the pairs themselves are arranged in a line, and the ordering of these pairs is significant. This is calculated by finding the factorial of the number of pairs, as each distinct ordering counts as a unique arrangement.

Bijections in Pairing

Establishing a one?to?one correspondence (or bijection) between two equally sized sets is a key concept in pairing problems. Here, matching each male dancer with a unique female dancer is achieved in factorial number of ways (specifically, n! for n pairs), with each assignment representing a distinct pairing configuration.

Transcript

Please give Ace some feedback