Problem 39 - Algebra for College Students

Question

In how many ways can six people be divided into two groups so that there are four in one group and two in the other? In how many ways can six people be divided into two groups of three each? \(15 ; 20\)

Step-by-Step Solution

Answer

15 ways for groups of 4 and 2; 10 ways for two groups of 3.

Step 1: Understanding the Problem for Group Sizes 4 and 2

We need to divide six people into two groups: one group with four people, and another group with two people. We need to find the number of ways we can form these groups.

Step 2: Choosing 4 People for the First Group

To form a group of four people from six, we choose any four out of the six. This is a combination problem, which can be solved using the combination formula: \[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15 \] Thus, there are 15 ways to choose four people for the first group.

Step 3: Automatic Group Formation for Remaining People

Once we have chosen four people for the first group, the remaining two people automatically form the second group. Thus, the number of ways to split the groups is just the number of ways to choose the first group, which is 15.

Step 4: Understanding the Problem for Two Groups of Three Each

Now we solve for splitting six people into two groups of three each. This presents a different scenario, as it's a balanced division.

Step 5: Choosing 3 People for the First Group

To form the first group of three people, we again use the combination formula:\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \] So, there are 20 ways to choose three people for the first group.

Step 6: Accounting for Identical Groups

However, since forming two groups of three is symmetrical (group 1 and group 2 are indistinguishable), we must account for this by dividing by 2 (to avoid double-counting).\[ \frac{20}{2} = 10 \] Thus, there are 10 distinct ways to divide the six people into two indistinguishable groups of three each.

Key Concepts

Combination Formula

In combinatorics, the combination formula is a key tool used when we need to select items from a larger set without regard to the order of selection. For choosing a subset of items, the formula is given by:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]Where:

• \(n\) is the total number of items,
• \(r\) is the number of items to choose,
• \(n!\) denotes the factorial of \(n\).

This formula, often read as 'n choose r', tells us the number of ways to choose \(r\) items from \(n\) options. Importantly, order doesn't matter in combinations, unlike permutations, where order is crucial.
In the exercise, we utilize this formula to calculate the number of ways to form groups. For example, when creating a group of four people from six, we use \(\binom{6}{4}\), resulting in 15 ways. This tells us there are 15 different sets of four people we can choose from the total of six.

Group Division

Group division is an interesting challenge in combinatorics. It involves splitting a set of items (or people, in our exercise) into smaller groups. Each group must satisfy certain conditions, such as size. In our example, we are dividing six people either into a group of four and a group of two, or two equal groups of three each.
When dividing into a group of four and a group of two, you first decide on the group of four using combinations. Once these four are chosen, the remaining automatically form the other group. Thus, the number of group formations is the same as the number of ways to choose the first group.
For equal groups, like the two groups of three, the division becomes slightly different. Since the groups are identical in size and indistinguishable, you must account for symmetry by dividing the total combinations by 2, which adjusts for the fact that swapping the groups doesn't result in a new division.

Symmetric Groups

Symmetry in group partitioning arises when groups are indistinguishable, as seen in our exercise where we divide six people into two groups of three. Here, each group of three is symmetrical with the other.
This symmetry means that a group of three chosen first can be the same as the one chosen later if only names are swapped. Therefore, to avoid counting these scenarios multiple times, we divide our total number of combinations by the number of equivalent scenarios:

• For two identical groups, the division factor is 2.

This factor correction (or symmetry consideration) ensures that the calculated result accurately reflects unique, distinguishable groups rather than permutations of the same configuration. Using symmetry help reflect real-life decision-making where only distinct variations matter.