Alicia lives in a town whose streets are on a grid system, with all streets running east - west or north - south without breaks. Her school, located on a corner, lies three blocks south and three blocks east of her hoem, also located on a corner. If Alicia only walks wouth or east on her way to school, how many possible routes can she take to school?

Answer

To find the number of possible routes Alicia can take to school, we can use combinatorial methods. Here’s a step-by-step solution:

### Step 1: Understand the Problem
Alicia needs to walk a total of 6 blocks to reach her school: 3 blocks south and 3 blocks east. Therefore, her journey consists of 3 south (S) moves and 3 east (E) moves.

### Step 2: Determine the Total Moves
The total number of moves Alicia will make is:
- South moves (S): 3
- East moves (E): 3
- Total moves = 3 + 3 = 6

### Step 3: Calculate the Combinations
The number of different routes can be calculated using the formula for combinations, which is given by:
\[ \text{Number of routes} = \frac{n!}{r_1! \cdot r_2!} \]
where:
- \( n \) is the total number of moves (6 in this case),
- \( r_1 \) is the number of south moves (3),
- \( r_2 \) is the number of east moves (3).

### Step 4: Substitute the Values
Substituting the values into the formula:
\[ \text{Number of routes} = \frac{6!}{3! \cdot 3!} \]

### Step 5: Calculate Factorials
Now, we calculate the factorials:
- \( 6! = 720 \)
- \( 3! = 6 \)

### Step 6: Substitute Factorials into the Equation
Now substitute the factorial values back into the equation:
\[ \text{Number of routes} = \frac{720}{6 \cdot 6} = \frac{720}{36} \]

### Step 7: Perform the Division
Now perform the division:
\[ \frac{720}{36} = 20 \]

### Conclusion
Thus, the total number of possible routes Alicia can take to school is **20**.