SUMMARY
The problem involves forming pairs from 10 women and 12 men, specifically creating 5 pairs consisting of one man and one woman. The correct approach to calculate the number of possible pairings is to use the formula #{12 \choose 5} \times {10 \choose 5} \times 5!#. This accounts for selecting 5 men and 5 women, followed by arranging them in pairs. The multiplication by 5! represents the different ways to order the selected pairs.
PREREQUISITES
- Understanding of binomial coefficients, specifically #{n \choose k#}
- Knowledge of factorials, particularly the concept of n!
- Basic combinatorial principles for pairing and ordering
- Familiarity with mathematical notation and problem-solving strategies in combinatorics
NEXT STEPS
- Study the concept of binomial coefficients in depth, focusing on their applications in combinatorial problems
- Learn about permutations and combinations, particularly how to calculate arrangements of selected items
- Explore advanced combinatorial techniques, such as the inclusion-exclusion principle
- Practice solving similar pairing problems to reinforce understanding of the principles discussed
USEFUL FOR
Students studying combinatorics, educators teaching mathematical concepts, and anyone interested in solving complex pairing problems in mathematics.