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Chapter 5: Problem 53

A family has six children. If this family has exactly two boys, how many different birth and gender orders are possible?

Short Answer

Expert verified
15

Step by step solution

01

Understand the Problem

There are six children in a family, and out of these six children, exactly two are boys. The task is to determine how many different birth and gender orders are possible.
02

Identify the Total Number of Positions

There are 6 positions because there are 6 children. We need to choose 2 out of these 6 positions for the boys.
03

Calculate the Number of Ways to Choose Positions for Boys

The number of ways to choose 2 positions out of 6 is given by the binomial coefficient formula:\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} \]
04

Simplify the Binomial Coefficient

Calculate the factorial values:\[ 6! = 720,\ 2! = 2,\ 4! = 24 \]Now simplify:\[ \binom{6}{2} = \frac{720}{2 \times 24} = \frac{720}{48} = 15 \]
05

Determine the Total Number of Birth and Gender Orders

Each of the selected positions for the boys can independently be filled as a boy or a girl. For the remaining 4 positions, they must be girls. The total number of different birth and gender orders is the binomial coefficient calculated in the previous step.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics focused on counting combinations and arrangements. It helps us answer questions like 'How many ways can we organize a certain set of items?' In this exercise, we use combinatorics to determine how many different birth and gender orders are possible when a family has six children with exactly two boys. By creating all possible arrangements, we can identify all scenarios in which the two boys can be placed among the six children.
Factorials
When solving problems involving combinatorics, we often encounter factorials. A factorial, denoted by an exclamation point (!), represents the product of all positive integers up to a given number. For example:
  • 3! = 3 脳 2 脳 1 = 6
  • 4! = 4 脳 3 脳 2 脳 1 = 24
In our exercise, we use factorials to calculate the binomial coefficient \( \binom{6}{2} \). This coefficient is found by dividing the factorial of the total number of children (6!) by the product of the factorial of the number of boys (2!) and the factorial of the remaining children (4!). This calculation gives us the number of ways to choose 2 boys out of 6 children.
Birth Order Probability
Birth order probability deals with the likelihood of different sequences of children's genders. In the family with six children, knowing that there are exactly two boys gives us a specific condition. We count how many orders meet this condition by using the binomial coefficient formula. Each unique set of positions for the boys is an order. In this exercise, the number of different birth and gender orders possible under these conditions is 15. This result is achieved by understanding how combinatorics allows us to count all possible placements of the boys among the six positions.

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Most popular questions from this chapter

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