14.3 - Casework

Links and useful resources

Concept summary and lesson

Consider different cases in order to count complex tings

So far, we've been counting everything by coming up with a single, grand formula and evaluating it. Sometimes that's not really practical (most of the time, really). In those cases, we'll need to count up parts of the problem separately and then add up the total afterwards.

Whenever you come across a counting problem that has choices that end up with different values, just use your knowledge to count those cases individually and then add the results.

Guided practice

The figure to the right represents a road map between 4 villages, labeled $A$ , $B$ , $C$ and $D$ . The arcs indicate paths between the various villages. How many ways are there to go from $A$ to $D$ along the paths, if you can only move left to right? (For example, you can’t go $A B A B A C D$ ; you can only go $A C D$ or $A B D$ .)
On the island of Abcdef, the alphabet has only 6 letters, and every word in their language has no more than 3 letters in it. How many words are possible? (A word can use a letter more than once, but 0 letters does not count as a word.)
How many pairs of positive integers $(a, b)$ satisfy $a^{2} + b < 24$ ?
A digital clock shows time in the form HH:MM. On a certain day, what is the number of minutes between 7:59 AM and 2:59 PM that HH is greater than MM?

Homework

#hw (math) (gr7) 14.3 Casework (p. 525) 2025-04-16