Multiplication principle

It is also called the fundamental principle of counting. It is the first tool in counting, which the Olympiad syllabus calls basic enumeration. It works for any number of steps. Multiply the number of ways at each step, as long as each step's count does not depend on earlier picks.

A small example

• You have 3 shirts and 4 pairs of trousers. For each shirt there are 4 trousers. So there are 3 × 4 = 12 outfits.
• How many two-digit codes can you make from the digits 1 to 9 if no digit repeats? The first digit has 9 choices. The second has 8, since one digit is used. So there are 9 × 8 = 72 codes.
• In how many orders can 4 books stand on a shelf? The first place has 4 choices, then 3, then 2, then 1. That is 4 × 3 × 2 × 1 = 24 orders.

The last answer is written 4! and read 4 factorial. Counting orders like this gives permutations. Counting choices where order does not matter gives the binomial coefficient.

Watch for the word or. If you choose a shirt or a pair of trousers, you add: 3 + 4 = 7 ways. The word and means multiply. When two groups overlap, adding counts some things twice. The inclusion-exclusion principle fixes that.

Where it fits

This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under combinatorics, as basic enumeration. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.

About the author

Mohit Sardana

Chief Architect

An engineer (B.Tech, Computer Science) with over 20 years of preparing students for entrance exams, including as Dean at FIITJEE and Vice-President at Aakash. He is the chief architect of the TRUpreBoards evaluation engine.

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