Question

Given \(1,2, \ldots, 11,\) select a subset of five elements from this set and a second subset with two of these elements. In how many ways can these groups be formed if: (a) There are no restrictions. (b) Each group contains all even or all odd integers. (c) No repetitions are allowed, and the smallest member of the second group is larger than the largest member of the first group. Show that it does not matter whether the two-element set or the five-element set is chosen first.

Short answer

(a) 4620; (b) 70; (c) 330.

Step-by-step solution

Understanding the Total Number of Ways

Firstly, for part (a), we need to select 5 elements from a set of 11. The number of ways to choose a subset of 5 elements from 11 is given by the binomial coefficient: \(\binom{11}{5} = 462.\)Next, we select 2 elements from the five chosen, which can be done in:\(\binom{5}{2} = 10.\)Therefore, the total number of ways to form both groups is:\(462 \times 10 = 4620.\)

Applying Restrictions For Even or Odd Sets

For part (b), we find sets where both groups are entirely even or entirely odd numbers. From 1 to 11, there are 5 even numbers (2, 4, 6, 8, 10) and 6 odd numbers (1, 3, 5, 7, 9, 11).For even elements, choose 5 out of 5:\(\binom{5}{5} = 1.\)For odd elements, choose 5 out of 6:\(\binom{6}{5} = 6.\)Once you have a 5-element set, choose 2 from it:\(\binom{5}{2} = 10.\)Thus, total if even: \(1 \times 10 = 10.\)Total if odd: \(6 \times 10 = 60.\)Hence, total ways: \(10 + 60 = 70.\)

Ensuring No Repetitions and Order

For part (c), we ensure each element in the 2-element set is larger than each in the 5-element set. First, choose a 7-element combination from the full set of 11 (since two sets combined must cover these without overlap):\(\binom{11}{7} = 330.\)Select a 2-element set from these 7:\(\binom{7}{2} = 21.\)The remaining 5 elements automatically become the 5-element set (as they're smaller). Thus, combinations:\(330.\) Inverting the process (choosing the 5 first then the 2), gives the same result due to symmetry in selection.

Key concepts

Set Theory

Set Theory is a fundamental concept in mathematics that deals with collections of objects, known as sets. A set is essentially a collection of distinct elements or members. In the context of this exercise, we are dealing with a set composed of numbers from 1 to 11.

Set operations often involve creating subsets, which are smaller collections of these elements. Subsets can vary in size and can either include certain members of the original set or none at all, which is termed as the empty set. When organizing or evaluating different configurations of subsets, set theory provides a foundational framework to understand the logic and relationships between various groups of elements.

By utilizing concepts from set theory, it is easier to visualize and comprehend complex problems involving selection and arrangement of different subsets within a larger set.

Binomial Coefficient

The Binomial Coefficient is a crucial tool in combinatorics, particularly useful for problems involving selection of items from a collection. It is denoted as \( \binom{n}{k} \), and it represents the number of ways to choose \( k \) elements from a larger set of \( n \) elements.

Mathematically, it is calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
This expression works by dividing the total permutations of \( n \) elements by the permutations of the chosen \( k \) elements and the remaining \( n-k \) elements.

In our exercise, the use of the binomial coefficient helps in determining the different ways we can select subsets. For instance, choosing 5 elements out of 11 is calculated using \( \binom{11}{5} \), and further selecting 2 elements from these 5 is \( \binom{5}{2} \). The concept simplifies complex counting tasks into manageable calculations.

Subset Selection

Subset Selection is an essential part of combinatorial mathematics. It involves selecting certain elements from a larger set to create a smaller group, known as a subset. This is particularly helpful in problems where you need to partition sets based on specific rules or restrictions.

For example, in the exercise, we have different subset selection conditions:

• Choosing a five-element subset and then a two-element subset without any restrictions.
• Creating subsets containing only even or only odd numbers.
• Forming subsets where every element in the smaller set is larger than the greatest element in the larger set.

Each of these requires a strategic approach to subset selection, ensuring all conditions are met. The flexibility and adaptability of subset selection make it a powerful tool for solving a variety of combinatorial problems. Understanding this concept can greatly simplify the process of solving complex mathematical tasks involving sets.