Math Problem Statement

The path from city M to city N can be taken by 6 paths, from city M to city O can be taken by 5 paths, from city N or city O to city P can be taken by 5 paths, while city N to city O can only be taken by 1 path. How many paths are there from city M to city O....

a. 100 ways

b. 110 ways

c. 120 ways

d. 130 ways

Solution

To solve this, we need to count the number of possible paths from City M to City O through City N or directly.

Here’s the breakdown:

  1. Direct paths from City M to City O:

    • There are 5 direct paths from City M to City O.
  2. Paths from City M to City O via City N:

    • First, there are 6 paths from City M to City N.
    • From City N, there is 1 path to City O.
    • Therefore, the number of ways to go from City M to City O via City N is: 6×1=6 paths6 \times 1 = 6 \text{ paths}

Total number of paths:

  • Direct paths from City M to City O = 5
  • Paths via City N = 6

Thus, the total number of paths from City M to City O is: 5+6=115 + 6 = 11

Extending the paths from City M to City P:

Now we need to add the possibilities from City O or City N to City P. However, since the problem only asks about the number of paths from City M to City O, the correct answer is:

11\boxed{11}

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Path Counting
Basic Multiplication

Formulas

Number of paths = direct paths + paths via another city

Theorems

Addition principle of counting

Suitable Grade Level

Grades 6-8