Counting with restrictions

Count each admissible outcome once. Decide whether order matters, whether a choice can be repeated, and whether cases overlap before selecting a formula. **Restriction** | **Strategy** | **Example** Repeated choices permitted | Multiply choices for each position. | Four-digit codes from 0–9, allowing leading zero: 104. Two distinct people together | Treat the pair as a block, then arrange inside it. | Six distinct people in a row: 5! x 2!. At least one chosen | Subtract the case with none. | Three people from four tutors and five students, with a tutor: 93- 53=74. Example Choose two students from five and one tutor from four: 52 41=40. Giving those three people distinct roles would add an arrangement step; a committee has no such step. Use disjoint cases for “exactly”, “at most” or “at least” restrictions. When using a complement, count the total and excluded outcomes in the same sample space.