46
Practical problems with the multiplication rule of combinatorics
Textbook, pp. 207–208
GoalKnow the multiplication rule of combinatorics and use it in practical problems.
New words
multiplication rule · ko‘paytirish qoidasiway of choosing · tanlash usulirepetition of digits · raqamlar takrorlanishicode · kod
Explanation
If the first choice can be made in m ways and then the second choice in n ways, the total number of ways to make both choices one after another is m · n. This is the multiplication rule; it works in “both ... and” situations. For example, choosing an outfit from 3 shirts and 4 pairs of trousers can be done in 3 · 4 = 12 ways: 4 trousers with each shirt. When making numbers from digits it matters whether repetition is allowed: if so, each place has the same number of options; if not, the number of options drops by one each time.
Worked examples
Two-digit numbers from the digits 1, 2, 3 (repetition allowed): 3 options for tens and 3 for ones, 3 · 3 = 9. Without repetition: 3 · 2 = 6.
A phone number begins with 224 and then has 4 arbitrary digits: 10 · 10 · 10 · 10 = 10,000 different numbers.
Class activity
“Choose an outfit”: pupils get 3 shirt cards and 4 trouser cards and make all outfits in a table or a tree diagram. They compare the count 12 with the multiplication rule.
Practice
1
How many outfits can be made from 3 shirts and 4 pairs of trousers?
12
2
How many three-digit numbers can be made from the digits 1, 2, 3, 4, 5 if digits may repeat?
125
3
How many three-digit numbers with no repeated digits can be made from 1, 2, 3, 4?
24
4
Why do we multiply rather than add the ways when choosing a shirt and trousers?
Each shirt can be worn with every pair of trousers, so the choices are made together: 3 groups with 4 options each.