2.1 Rule of Product

2.1 Rule of Product
2.2 Rule of Sum

4.1 Permutation when taken all at a time
4.2 Permutation when taken $r$ at a time

Statement: If one experiment has $n$ possible outcomes and another experiment has $m$ possible outcomes, then there are '$m \times n$' possible outcomes when both the experiments are performed.

Proof: Let there be be two events say $M$ and $N$,

Let the outcomes of $M$ be, $${a_1},{a_2},........,{a_m}$$

and the outcomes of $N$ be, $${b_1},{b_2},........,{b_n}$$

Then one can choose the following way,

$$\{ {a_1}{b_1},{a_2}{b_1},.......,{a_m}{b_1}\} ,\{ {a_1}{b_2},{a_2}{b_2},.......,{a_m}{b_2}\} ,.....,\{ {a_1}{b_n},{a_2}{b_n},.......,{a_m}{b_n}\} $$

Here each set has '$m$' elements and there are '$n$' sets in total.

Thus the total number of elements is '$m \times n$ '.

Question 1. In a juice shop there are two categories namely milkshakes and ice-creams. There are $5$ flavours of milkshake and $7$ flavours of ice-cream. Find the possible number of choices you have in case you wish to have both milkshake and ice-cream.

Solution: There are $5$ flavours of milkshake, thus there are $5$ ways of choosing one.

There are $7$ flavours of ice-cream, thus there are $7$ ways of choosing one.

Thus the total number of choices one has to choose one from each category is, $$=5 \times 7=35$$

Note: The product rule can be applied for any number of events occurring together.

Question 2. For a class committee meeting in a school, the pupil leader has to choose one from each of the three sections. The section $A$ has selected $5$, section $B$ has selected $3$ and section $C$ has selected $4$ as representatives, from which the pupil leader selects one from each. Find the number of possible choices he has in choosing the committee.

Solution: Possible number of ways,

From Section $A = 5$ ways

From Section $B = 3$ ways

From Section $C = 4$ ways

Thus the total number of choices is, $$=5 \times 3 \times 4 = 60$$