Problem 66 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Choosing a Group Sixteen boys and nine girls go on a camping trip. In how many ways can a group of six be selected to gather firewood, given the following conditions? (a) The group consists of two girls and four boys. (b) The group contains at least two girls.

Step-by-Step Solution

Answer

(a) 65520 ways; (b) 127680 ways.

Step 1: Understanding the Problem

We need to form a group of six people consisting of sixteen boys and nine girls. We apply combinations to determine different group formations: (a) exactly 2 girls and 4 boys, (b) at least 2 girls.

Step 1: Calculate Ways for (a)

To select a group with 2 girls and 4 boys, we determine the number of ways to select 2 girls from 9 and 4 boys from 16. Use the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). First, calculate for girls: \( \binom{9}{2} = \frac{9!}{2!(9-2)!} = 36 \). Then, calculate for boys: \( \binom{16}{4} = \frac{16!}{4!(16-4)!} = 1820 \). The total number of ways is the product of these combinations: \( 36 \times 1820 = 65520 \).

Step 2: Calculate Ways for (b) with Two Girls

For at least two girls, start by calculating scenarios with exactly 2 girls (already calculated in (a)). Thus, we have 65520 ways as concluded in Step 1.

Step 3: Calculate Ways for (b) with Three Girls

Calculate the groups with 3 girls and 3 boys. Use combinations for three girls from nine: \( \binom{9}{3} = 84 \). And for boys: \( \binom{16}{3} = 560 \). Thus, the number of ways is \( 84 \times 560 = 47040 \).

Step 4: Calculate Ways for (b) with Four Girls

Calculate the groups with 4 girls and 2 boys. For selecting 4 girls from 9: \( \binom{9}{4} = 126 \). For selecting 2 boys from 16: \( \binom{16}{2} = 120 \). Thus, the number of ways is \( 126 \times 120 = 15120 \).

Step 5: Calculate Total Ways for (b)

To find the total number of ways the group contains at least 2 girls, sum up the number of ways with 2 girls, 3 girls, and 4 girls. This is calculated as \( 65520 + 47040 + 15120 = 127680 \).

Step 7: Final Answer

The solution to (a) is: 65520 ways. The solution to (b) is: 127680 ways.

Key Concepts

Combinations

When we talk about combinations in mathematics, we're exploring how to select items from a group, where the order doesn't matter. This concept is crucial in many real-world scenarios, like forming teams or selecting group members.
The formula for combinations is given by \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to select. The exclamation point signifies a factorial, which means you multiply a series of descending natural numbers. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).
In our exercise, the goal was to choose a group of people for specific tasks. We used combinations to determine how many different groups could be formed. For instance, to find how many ways we can choose 2 girls out of 9, we utilize the combination formula to calculate \( \binom{9}{2} \). This helps us figure out the possible selections without considering the order.

Permutations

Permutations consider the arrangement of items where order does matter. This concept is different from combinations because you have to account for every possible sequence.
The formula for permutations is \( P(n, r) = \frac{n!}{(n-r)!} \). This formula calculates all possible arrangements of \( r \) items from a set of \( n \).
Although our exercise primarily involves combinations, understanding permutations broadens your grasp of counting principles. If order were important in choosing the group of campers, then we'd employ permutations. For example, if we selected leaders among the group of campers, each unique arrangement would offer a new scenario, making permutations essential.

Counting Principles

Counting principles serve as foundational techniques in determining the number of ways events can occur. The most common principles include the addition and multiplication principles.

• Addition Principle: If one event can occur in \( m \) ways and a second event can occur in \( n \) ways, and if these events cannot occur at the same time, then there are \( m + n \) ways for either event to occur.
• Multiplication Principle: If one event can occur in \( m \) ways and another independent event can occur in \( n \) ways, the two events can occur together in \( m \times n \) ways.

In solving our exercise, the multiplication principle was used to find the number of ways to select a group of campers. By calculating the ways to select girls and then boys separately, we use the multiplication rule to find the total number of ways to form the group.

Algebra Problem Solving

Algebra problem solving involves the use of algebraic methods to find unknown values in equations or problems. It requires converting verbal descriptions and conditions into mathematical expressions.
In our current context, algebra underpins the use of formulas such as combinations to calculate possible selections and derive solutions. By employing algebra, we can systematically approach the problem, ensuring all criteria are considered.
One approach in algebra is identifying expressions that represent the problem components, like representing girls or boys using combination expressions. In cases where problems involve various scenarios, such as having "at least" or "at most" constraints, algebra helps to organize these cases and sum different possibilities accurately, as demonstrated when finding selections that include at least two girls.