32
The basic rule of combinatorics and its use in problems
Textbook: pp. 203–209
GoalKnow the multiplication and addition rules and use them to solve combinatorial problems.
New words
multiplication rule · ko‘paytirish qoidasiaddition rule · qo‘shish qoidasistage of choice · tanlash bosqichidistinct digits · takrorlanmaydigan raqamlar
Explanation
Multiplication rule (the basic rule of combinatorics): if the first choice can be made in m ways and the second (after each first choice) in n ways, then the two choices in succession can be made in m · n ways. The rule extends to more than two stages: multiply the numbers of ways. Addition rule: if an object can be chosen in m ways or from another group in n ways, and the groups do not overlap, there are m + n ways in all. For numbers with distinct digits, each following place has one fewer possibility; also remember that 0 cannot stand first.
Worked examples
A pupil chooses one of 3 pens and one of 4 notebooks: 3 · 4 = 12 sets. 3 shirts and 2 pairs of trousers: 3 · 2 = 6 outfits.
Three-digit numbers with distinct digits: the first digit has 9 choices (not 0), the second 9 (0 is allowed but not the first digit), the third 8: 9 · 9 · 8 = 648. Choosing one club from 3 sports clubs or 4 art clubs: 3 + 4 = 7 ways (addition rule).
Class activity
“Combination lock”: pupils guess the number of possibilities for a 3-digit lock code, then compute it with the multiplication rule to check the guess.
Practice
1
A menu has 4 soups and 5 main courses. How many different lunches with one soup and one main course are there?
20
2
How many three-digit codes can be made from the digits 1–5 (repetition allowed)?
125
3
How many four-digit numbers with distinct digits are there?
4536
4
Why are the numbers of ways multiplied, not added, for successive choices?
Because every way of the first choice goes with all ways of the second, so each of the m ways is repeated n times.