a child has five coloured blocks: one red, one blue, one green, one yellow and one white. she stacks two, three, four or five different blocks on top of one another to form a vertical tower. how many different towers are there that she could form that are three blocks high?

To solve this problem, we need to find the number of different towers of height three that can be formed using five different colored blocks: red, blue, green, yellow, and white.

### Step-by-step Solution:

1. **Identify the Key Elements:**
   - We have 5 different colored blocks: red (R), blue (B), green (G), yellow (Y), and white (W).
   - A tower is 3 blocks high, meaning each tower will consist of 3 out of the 5 available blocks.
   - The order in which the blocks are stacked matters (since it's a tower).

2. **Calculate the Number of Ways to Choose 3 Blocks Out of 5:**
   - First, we choose 3 blocks out of the 5 available blocks. The number of ways to do this is given by the combination formula:
   \[
   \binom{5}{3} = \frac{5!}{3! \cdot (5-3)!} = \frac{5 \cdot 4}{2 \cdot 1} = 10
   \]

3. **Calculate the Number of Permutations for Each Combination:**
   - For each combination of 3 blocks, the blocks can be arranged in any order. The number of permutations of 3 blocks is:
   \[
   3! = 3 \times 2 \times 1 = 6
   \]

4. **Calculate the Total Number of Towers:**
   - The total number of different towers that are 3 blocks high is the product of the number of combinations and the number of permutations for each combination:
   \[
   \text{Total Towers} = \binom{5}{3} \times 3! = 10 \times 6 = 60
   \]

### Conclusion:

There are **60 different towers** that the child could form that are three blocks high.

Would you like more details or have any questions? Here are some related questions:

1. How many different towers can be formed if the tower must be exactly four blocks high?
2. What if the order of the blocks in the tower did not matter? How would the calculation change?
3. How many towers can be formed if at most five blocks can be used?
4. If the blocks were identical except for their color, how would that affect the number of combinations?
5. What is the probability of randomly forming a tower that is exactly three blocks high and includes a red block?

**Tip:** Understanding permutations and combinations is key to solving problems involving arrangements where order matters.

A child has five coloured blocks: one red, one blue, one green, one yellow and one white. She stacks two, three, four or five different blocks on top of one another to form a vertical tower. How many different towers are there that she could form that are three blocks high?

Learn how to calculate the number of different towers that can be formed by stacking three blocks from five colored blocks. This combinatorics problem teaches key concepts like combinations and permutations, suitable for students in Grades 6-8.

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