Problem 47 - Algebra 2 and Trigonometry

Question

There are 12 boys and 13 girls assigned to a class. In how many ways can 2 boys and 3 girls be selected to transfer to a different class?

Step-by-Step Solution

Answer

18876 ways.

Step 1: Understanding the Problem

You need to find the number of ways to select a group of students. Specifically, you are selecting 2 boys from a group of 12 boys and 3 girls from a group of 13 girls.

Step 2: Choosing 2 Boys

The number of ways to select 2 boys from 12 is given by the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Thus, the number of ways to choose 2 boys from 12 is \( \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \).

Step 3: Choosing 3 Girls

Similarly, to find the number of ways to select 3 girls from 13, use the combination formula: \( \binom{13}{3} = \frac{13!}{3!(13-3)!} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \).

Step 4: Calculating Total Combinations

The total number of ways to select 2 boys and 3 girls is the product of the individual combinations. Therefore, it is \( 66 \times 286 = 18876 \).

Key Concepts

Binomial Coefficient

The term **binomial coefficient** is a key concept in combinatorics, which helps in determining the number of ways to choose a subset of items from a larger set without considering the order of selection. It is represented by the symbol \( \binom{n}{r} \), which is read as "n choose r."

• "n" denotes the total number of items available.
• "r" represents the number of items to choose from that set.

For example, in our problem, to choose 2 boys out of 12, we use \( \binom{12}{2} \). The logic behind the binomial coefficient is to count all possible combinations, a crucial idea that forms the foundation of probability and statistics. The elegance of binomial coefficients is how they translate a complex counting problem into a simple mathematical expression.

Combination Formula

The **combination formula** is a powerful tool that helps manage selection without regard to order. The formula used to calculate a combination is:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here is how each component works:

• "n!" (n-factorial) calculates the total possible arrangements of the items.
• "r!" accounts for the arrangements of the chosen items.
• "(n-r)!" accounts for the arrangements of the remaining items.

Utilizing this formula allows us to determine the different ways of selecting "r" objects from a set of "n" objects. Apply the formula for choosing 2 boys from 12 in our given problem: \( \binom{12}{2} = \frac{12!}{2!(12-2)!} \). This organizes the counting process by simplifying permutations into combinations.

Factorial

**Factorial** is a mathematical operation essential for combinations and permutations. It is represented by the symbol "!" and means multiplying a series of descending natural numbers.

• "n!" denotes the product of all positive integers up to "n."

For example, calculating 4-factorial (4!) results in:\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]It simplifies calculations in combinatorics by efficiently managing large numbers. In the combination formula, factorials reduce the complexity by streamlining the multiplication of large sequences, such as breaking down \( 12! \) during our task of computing \( \binom{12}{2} \). Understanding factorials is vital for using formulas that depend on simplifying with factorial reductions to manage computational complexity.