Webinars SET B - Grade 9-10 - Sunday@6-7pm EST-Recording Webinar 6

Video Transcription

Video Summary

The lesson covers a range of combinatorial counting techniques, illustrated through various problem-solving examples. It begins with a warm-up on counting ways to choose distinct white and black squares from a checkerboard under certain constraints, using the multiplication principle. The agenda includes reviewing fundamental counting methods such as casework, constructive counting, complementary counting, and recursive counting.<br /><br />Casework involves dividing problems into distinct cases and summing their counts, while constructive counting explicitly builds the counted objects. Complementary counting simplifies problems by counting the complement, and recursive counting relates larger problems to smaller cases via recurrence relations.<br /><br />Examples include counting ten-digit numbers formed from digits 1, 2, and 3 where consecutive digits differ by exactly 1, illustrating constructive and casework strategies. Another problem considers path counting on triangular grids using recurrence relations reminiscent of Pascal’s triangle.<br /><br />The lesson explains permutations—arrangements where order matters—and combinations, where order does not matter, including adaptations for repeated elements. The classic factorial formulas are explored through problems featuring rings that must be removed under order constraints, and digit arrangements with restrictions on sums or divisibility.<br /><br />Advanced examples involve counting seven-digit numbers with repeated digits and identifying numbers whose digits sum to 10 from digits 1 and 3, broken down into cases and using permutation with repetition and combination principles.<br /><br />Probability is introduced as the ratio of favorable outcomes to total equally likely outcomes, demonstrated through the example of nine-digit numbers divisible by 18.<br /><br />Finally, a complex problem requires arranging digits 1 through 9 so that every three consecutive numbers sum to a multiple of three; this is analyzed by considering patterns of remainders modulo 3 and breaking the problem into manageable subproblems.<br /><br />The lesson emphasizes understanding underlying logic over memorizing formulas and advocates flexibility in applying multiple counting techniques when solving combinatorial problems.

Keywords

combinatorial counting

multiplication principle

casework

constructive counting

complementary counting

recursive counting

permutations

combinations

probability

modular arithmetic