The rule of sum and the rule of product
In how many ways can we choose "a or b" and "a and b": the sum rule, the product rule, pairs (a; b), a road scheme and problems about digits. This follows the Ukrainian Year 9 algebra book, section 20 "Basic rules of combinatorics".
- Ukraine: grade 9
Guide
In the Ukrainian Year 9 algebra book (section 20, "Basic rules of combinatorics") counting problems are solved by two rules. The sum rule. If a set A has m elements and a set B has k elements and they have no elements in common, then the choice "a or b", where a∈A, b∈B, can be made in m + k ways.
The product rule. If an element a can be chosen in m ways and after each such choice an element b can be chosen in k ways, then the choice "a and b" in that order can be made in m·k ways. All the pairs (a; b) can be written in a table: rows are the choice of a, columns the choice of b. For three choices the number of ways is m·k·n.
When a problem needs both rules, as in a road scheme from city A to city B through M or through N, first count each route with the product rule and then add with the sum rule.
The tree shows all the pairs (a; b): tap + to grow the branches. The other problems are solved without a picture. The book writes the answer like this: "Answer: 14 ways".