Multiplication principle

This sub-section states the Multiplication Principle formally: if one operation can be carried out in $m$ ways, followed by a second operation that can be carried out in $n$ ways, and the two operations are independent of each other, then the two operations TOGETHER can be carried out in $m\times n$ ways. Unlike the Addition Principle (a single choice made from exclusive alternatives), the Multiplication Principle applies when BOTH operations must be performed as part of the same overall task, one after the other.

Two worked examples anchor the principle. Example 1: Samadhan Bhojanalay's thali has four independently-chosen items — roti (2 choices: chapati, tandoor roti), rice (3 choices: plain rice, jeera rice, dal khichadi), vegetable (3 choices: dum aloo, paneer masala, mixed veg), and dal (2 choices: dal fry, dal tadka) — and since exactly one option is picked for each of the four items to make up a single thali, the total number of distinct thali menus is $2\times3\times3\times2=36$. Example 2: a company labels each product with a code of two letters followed by three digits; the two letters can be formed in $26\times26=676$ ways (since a letter can repeat) and the three digits in $10\times10\times10=1000$ ways, so by the Multiplication Principle the total number of distinct product labels is $676\times1000=6{,}76{,}000$.

Two Activities invite hands-on practice distinguishing this principle from the Addition Principle covered in the previous sub-section: Activity 1 asks how many ways Suresh can select ONE pencil (from 4) AND ONE eraser (from 2) together for his exam — a Multiplication Principle case, since both choices are made jointly ($4\times2=8$ ways). Activity 2 asks how many ways Sunil can select just ONE ballpen overall, from 4 ballpens of one company or 3 of another — this is actually an Addition Principle case in disguise (a single pen chosen from one exclusive group or the other, $4+3=7$ ways), deliberately placed right after Activity 1 so the reader has to actively decide, and give reasons, for which principle fits which scenario — reinforcing that the key diagnostic question is whether a SINGLE choice is being made from alternatives (add) or several choices are being made TOGETHER (multiply). …