Problem 30 - Calculus for Biology and Medicine

Question

Twelve children are divided up into three groups, of five, four, and three children, respectively. In how many ways can this be done if the order within each group is not important?

Step-by-Step Solution

Answer

There are 27,720 ways to divide the children into groups of 5, 4, and 3.

Step 1: Calculate Ways to Select the First Group

Since the order within the group is not important, we can use combinations to determine the number of ways to select the first group of 5 children from the total of 12. This is given by the combination formula: \( \binom{12}{5} = \frac{12!}{5!(12-5)!} \). Calculate this value to find the number of ways to select the first group.

Step 2: Calculate Ways to Select the Second Group

After selecting the first group of 5 children, 7 children remain. We need to form the second group with 4 of these remaining children. The number of ways to do this is \( \binom{7}{4} = \frac{7!}{4!(7-4)!} \). Calculate this value to find the number of ways to select the second group.

Step 3: Calculate Ways to Form the Last Group

Once the first two groups have been formed, the remaining 3 children automatically form the last group. Since there is only one way to choose all 3 remaining children, it is straightforwardly \( \binom{3}{3} = 1 \).

Step 4: Multiply to Find the Total Arrangements

Multiply the number of ways calculated in Steps 1, 2, and 3. This will give the total number of ways to divide the children into three groups of 5, 4, and 3 respectively. The total number of ways is given by \( \binom{12}{5} \times \binom{7}{4} \times \binom{3}{3} \).

Step 5: Evaluate the Numerical Result

Calculate each combination: \( \binom{12}{5} = 792 \), \( \binom{7}{4} = 35 \), and \( \binom{3}{3} = 1 \). Multiply these values together: \( 792 \times 35 \times 1 = 27720 \). Thus, there are 27,720 ways to divide the children into the specified groups.

Key Concepts

Permutations and Combinations

Understanding the difference between permutations and combinations is crucial when dealing with problems related to arranging or grouping objects. **Permutations** refer to the different arrangements of a set of items where the order matters. For example, arranging students in line involves permutations because the order in which they stand affects the outcome.

Conversely, **combinations** are used when the order does not matter. This is exactly what is used in our problem about dividing children into groups. The formula for combinations, which is denoted as \( \binom{n}{r} \), represents the number of ways to choose \( r \) items from \( n \) without regard to the order of selection. Mathematically, it is represented as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n! \) (n factorial) is the product of all positive integers less than or equal to \( n \).

In permutation problems, if each arrangement of items is distinct due to the order of selection, then permutations are used, often calculated with: \[ P(n, r) = \frac{n!}{(n-r)!} \]. Hence, choosing the right method between permutations and combinations depends on whether the order matters in the given scenario.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. It is a field that includes a variety of techniques used for calculating possible configurations of objects and is essential for understanding complex problems in fields like computer science, physics, and engineering.

In the context of our exercise, combinatorics helps in determining how many ways we can divide a set of twelve children into groups without considering any order within each group. We apply combinations to find the number of ways to select specific groups of children from the total set:

• First, choose a group of 5 from 12 children using \( \binom{12}{5} \).
• Then, choose 4 from the remaining 7 children using \( \binom{7}{4} \).
• The final remaining children automatically form the last group since there are no choices left, calculated as \( \binom{3}{3} \).

Combinatorics facilitates breaking down larger problems into smaller, more manageable tasks, which can then be methodically solved to achieve the desired outcome.

Mathematical Counting Principles

Mathematical Counting Principles are foundational concepts that guide us in determining the number of ways events can occur. They encompass rules and formulas that simplify the counting of combinations, permutations, and arrangements of sets.

In the given problem, the counting principle is applied through a step-by-step decomposition of the total grouping task:

• **Product Rule**: Multiply together the number of ways to accomplish each step in a multi-step procedure. For instance, once groups are formed step-by-step, multiplying these combinations \( \binom{12}{5} \times \binom{7}{4} \times \binom{3}{3} \) gives the total number of ways.
• **Sum Rule**: This would apply if there were different, non-overlapping processes contributing to a task.

By following these principles, we derive that there are 27,720 ways to group the children, as each decision is compounded by those following it, an effective demonstration of the multiplicative aspect of counting principles. Mastering these principles empowers students to tackle similarly structured problems with confidence.