Lesson Plans SET B for Grades 9-10-Lesson 6 Combinatorics HW

Lesson 6 Combinatorics HW

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This document contains a set of combinatorics practice problems aimed at students in levels 9-10, following a Math Kangaroo class. Each problem encourages application of combinatorial concepts covered in Lesson 6.<br /><br />1. <strong>Dice Rolling Probability:</strong> The problem involves rolling a fair 6-sided die 4 times and finding the probability of obtaining a sequence where each neighboring number has different parity (odd/even). By evaluating two possible alternating sequences (OEOE and EOEO), and acknowledging each number has a 50% odd/even probability, the total probability is calculated as \( \frac{1}{8} \).<br /><br />2. <strong>Letter Arrangement in 'MATHKANGAROO':</strong> Here, the task is to arrange the letters such that the two 'O's are not adjacent. Using factorial calculations, the solution involves finding total arrangements then subtracting those where the 'O's are together, resulting in \( \frac{11!}{3!} \).<br /><br />3. <strong>Children Lining Up:</strong> With 10 children and the stipulation that Sam must be in front of Lily, the problem calculates the number of valid permutations by fixing Sam's position in relation to Lily, resulting in \( \frac{10!}{2!} \) total arrangements.<br /><br />4. <strong>Stacked Gift Boxes Removal:</strong> This assesses how many ways the boxes can be removed respecting their stack order. The result is calculated as \( \frac{9!}{4! \cdot 3! \cdot 2!} \), considering each stack’s internal order.<br /><br />5. <strong>Triangulating a Hexagon:</strong> The method uses a recursive approach, referencing known triangulations for smaller polygons like quadrilaterals and pentagons to avoid overcounting. The number of ways to triangulate a hexagon is computed as 14.<br /><br />These problems enhance understanding of combinatorial reasoning, and the document details the solutions and reasoning, fostering comprehension through practice.

Keywords

combinatorics

practice problems

Math Kangaroo

probability

dice rolling

letter arrangement

permutations

triangulation

hexagon

combinatorial reasoning