Lesson 1 of 11 · 7 min

The multiplication principle

NCERT §6.1, §6.2

Imagine the school's sports day. Every runner gets a kit: a T-shirt in one of 3 house colours, one of 2 caps and one of 2 water bottles. How many different kits can the sports teacher hand out, without laying every one of them on a table?

The story this chapter follows: Sports day at school

Imagine a school sports day. Runners get a kit of one of 3 T-shirts, 2 caps and 2 water bottles; lockers have 3-dial locks; 12 students compete for captain and vice-captain; 4 of the 7 fastest sprinters run the relay; the house banner spells BALLOON; and a team of 5 is picked from 4 girls and 6 boys. The numbers are made up for easy arithmetic, and they carry every example in this chapter.

The lesson in notes

In short

Counting techniques find how many ways something can happen without writing out every possibility, which quickly becomes impossible as the numbers grow.

Multiplication principle: if one job can be done in m ways and, after it, a second job can be done in n ways, the two jobs in that order can be done in m × n ways. The rule extends to three or more jobs: m × n × p, and so on.

Sports-day kit: a runner picks one of 3 T-shirt colours, one of 2 caps and one of 2 water bottles. The number of different kits is 3 × 2 × 2 = 12, which a tree diagram with 3, then 6, then 12 branch ends confirms.

Locker lock: 3 dials, each showing a digit 0 to 9. If digits may repeat, there are 10 × 10 × 10 = 1000 codes. If no digit may repeat, the second dial has only 9 choices left and the third 8, giving 10 × 9 × 8 = 720 codes.

Fill the most restricted place first. For two-digit even numbers from the digits 1 to 6 with repetition allowed, the units place must be 2, 4 or 6 (3 ways) and the tens place can be any of the 6 digits, so there are 3 × 6 = 18 such numbers.

A coin tossed 4 times has 2 × 2 × 2 × 2 = 16 possible outcome sequences, because each toss has 2 results whatever happened before.