Question

Suppose there are three routes from North Point to Boulder Creek, four routes from Boulder Creek to Beaver Dam, four routes from Beaver Dam to Star Lake, and two routes directly from Boulder Creek to Star Lake. (Hint: Draw a sketch. Assume each route can pass through a city at most one time.) (a) How many routes from North Point to Star Lake pass through Beaver Dam? To answer this question, think of creating a route from North Point to Star Lake that passes through Beaver Dam as a 3-step process. Step 1: Choose a route from North Point to Boulder Creek . Step 2: Choose a route from Boulder Creek to Beaver Dam. Step 3: Choose a route from Beaver Dam to Star Lake. There are choices for step 1, choices for step 2, and choices for step 3. Thus, the answer is . (b) How many routes from North Point to Star Lake bypass Beaver Dam?

Answer 1

(a) 36 routes

(b) 6 routes

Explanation:

Given

`North\ Point \to Boulder\ Creek = 3`

`Boulder\ Creek \to Beaver\ Dam = 4`

`Beaver\ Dam \to Star\ Lake= 4`

`Boulder\ Creek\to Star\ Lake= 2`

Solving (a): From
`North\ Point` to
`Star\ Lake` through
`Beaver\ Dam`

To do this, we consider the following routes;

`North\ Point \to Boulder\ Creek \to Beaver\ Dam \to Star\ Lake`

Taking a pair of routes, we have:

`North\ Point \to Boulder\ Creek = 3`

`Boulder\ Creek \to Beaver\ Dam = 4`

`Beaver\ Dam \to Star\ Lake= 4`

By multiplication route, we have:

`Routes = 3 * 4 * 3`

`Routes = 36`

(b) From
`North\ Point` to
`Star\ Lake` bypass
`Beaver\ Dam`?

To do this, we consider the following routes

`North\ Point \to Boulder\ Creek \to Star\ Lake`

Taking a pair of routes, we have:

`North\ Point \to Boulder\ Creek = 3`

`Boulder\ Creek\to Star\ Lake= 2`

By multiplication rule, we have:

`Routes = 3 * 2`

`Routes = 6`