Sum and Product Rule

Four college students enter an elevator on the ground floor of a 6-story dormitory. If each student can independently get off on only different floors, in how many different ways can the four students exit the elevator ?

A dice is rolled 4 times. Find the number of rolls in which
(i) 6 appears on all 4 dices
(ii) 6 appears on none of the dices
(iii) 6 appears at least once

$$\begin{aligned} &\text { Let } x, y, z \in\{1,2,3, \ldots, 15\} \text {. Find the number of ordered triples }(x, y, z) \text { such that: }\\ &x \leq 4<y \leq 10<z \end{aligned}$$

There are 5 routes between Boston and New York, and 3 routes between New York and Philadelphia.

(i) In how many ways can someone drive from Boston to Philadelphia by way of New York?

(ii) In how many ways can someone drive from Boston to Philadelphia and back to Boston, passing through New York on both trips?

(iii) In how many ways can someone drive the round trip described in (ii) without using the same route more than once?

Four college students enter an elevator on the ground floor of a 6-story dormitory. If each student can independently get off on any of the 6 floors, in how many different ways can the four students exit the elevator ?

In how many ways can a student answer 4 multiple choice questions ? Each question has 3 choices.

A city has 10 gates. In how many ways can a person enter the city through one gate and come out from the other ?

There are 25 girls and 35 boys in a class.
(i) How many ways to pick one of the students as class representative.
(ii) How many ways to pick a boy representative and a girl representative

There are 25 gates to enter a city from north and 45 gates to enter a city from south.
In how many ways can a person enter the city ?

Out of 12 Geography, 46 Science, and 66 Calculus books, in how many ways can one book be selected ?