Subjects combinatorics

Boys Together

Step-by-step solutions with LaTeX - clean, fast, and student-friendly.

Search Solutions

Boys Together


1. **Problem statement:** We need to find the number of ways to seat 6 girls and 2 boys in a row such that the two boys are always seated together. 2. **Understanding the problem:** Treat the two boys as a single unit since they must sit together. This reduces the problem to arranging 7 units: 6 girls + 1 "boy pair". 3. **Formula and rules:** - Number of ways to arrange $n$ distinct objects in a row is $n!$. - Since the two boys can switch seats within their pair, multiply by $2!$ for their internal arrangements. 4. **Calculations:** - Arrange 7 units: $7!$ - Arrange 2 boys within their unit: $2!$ - Total arrangements: $7! \times 2!$ 5. **Evaluate:** - $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$ - $2! = 2$ - Total = $5040 \times 2 = 10080$ 6. **Answer:** There are **10080** ways to seat 6 girls and 2 boys in a row with the boys together.