Problem 14 - Algebra and Trigonometry

Question

Birth Order In a family of four children, how many different boy-girl birth- order combinations are possible? (The birth orders \(B B B G\) and \(B B G B\) are different.)

Step-by-Step Solution

Answer

There are 16 possible birth-order combinations of boys and girls.

Step 1: Understanding the Problem

In this exercise, we need to calculate how many different birth order combinations of boys (B) and girls (G) are possible in a family with four children. Each birth order sequence is unique if the arrangement of B's and G's is different. For example, BBBG is different from BBGB even though they both contain three Boys and one Girl.

Step 2: Formulating the Combination

For four positions where each can be either a boy (B) or a girl (G), we are essentially finding permutations of B's and G's. If there are no restrictions, each position has 2 possibilities: a boy or a girl, leading to an exponential function of possible combinations.

Step 3: Calculating Total Combinations

Using the formula for binary combinations, the total number of possible combinations is given by the number 2 raised to the power of the number of children: \[ 2^4 = 16 \]Hence, there are 16 different possible combinations of birth order.

Key Concepts

Birth Order Combinations

When thinking about birth order combinations in a family of four, it's all about how the boys and girls can be arranged. Imagine that each of the four children can either be a Boy (B) or a Girl (G). Each distinct arrangement, like having all boys followed by a girl or alternating boy-girl sequences, counts as a separate combination. The vital idea here is that any sequence of birth order, such as 'BBBG' or 'BGBG', is unique if the positions of the boys and girls differ. This means every new arrangement offers a new combination.
Imagine it like a puzzle where the order in which you place pieces matters.

• The sequence 'BBBG' is not the same as 'BBGB'.
• Although they both have three boys and one girl, the specific order gives them individuality.

Understanding this concept is crucial for grasping how permutations in sequences, like birth order, differ and why they each represent a unique possibility.

Binary Combinations

Binary combinations involve choices where there are only two possibilities for each choice. In our context of birth order, each position in the sequence can either be a Boy (B) or a Girl (G). This setup is comparable to a simple binary system comprising of two options—much like flipping a coin where you can get heads or tails.The formula used to determine the total possible combinations of birth orders is exponential: \[2^n\]where \(n\) represents the number of events—in this case, the number of children.
For a family of four, it translates to:

• Each child has 2 possible outcomes (B or G).
• Thus, the total possible combinations is \(2^4 = 16\).

Every combination represents a possible birth order sequence by selecting one of the two choices for each child. This method of working through all binary possibilities ensures that all sequences are accounted for.

Permutations in Probability

Permutations are all about the arrangement or order of items. When you're dealing with permutations in probability, you're examining the various ways you can order a set number of elements. In terms of birth order with four children, permutations consider each different arrangement of boys and girls as separate outcomes. Hence, the set is comprised of different sequences where even a simple swap in placement results in a new permutation.
Let's delve into why the order is so significant:

• While the number of each gender matters, it's their placement that defines permutations.
• Hence, 'BBGB' is a distinct permutation when compared to 'BGBB'.

Unlike simple combinations, permutations rely on precise positions, showcasing how many different ordered sequences are possible from the same set of elements. This highlights the potential number of outcomes when considering sequences with specific orderings in probabilistic settings.