You are given a sequence of 15 different tasks. In how many ways can you select 7 tasks such that no two selected tasks are adjacent in the sequence?

To solve the problem of selecting 7 tasks from a sequence of 15 different tasks such that no two selected tasks are adjacent, we'll use a combinatorial approach.

### Step 1: Visualizing the Selection

Consider the sequence of 15 tasks as:

\[
\text{T}_1, \text{T}_2, \ldots, \text{T}_{15}
\]

We need to select 7 tasks, ensuring that no two of the selected tasks are adjacent.

### Step 2: Introduce Gaps

If we select 7 tasks, there will be \( 15 - 7 = 8 \) tasks that are not selected. These unselected tasks will create gaps between the selected tasks. For example, if we have the selected tasks as \(\text{T}_{a_1}, \text{T}_{a_2}, \ldots, \text{T}_{a_7}\), then the gaps between them can be represented as follows:

\[
\_ \quad \text{T}_{a_1} \quad \_ \quad \text{T}_{a_2} \quad \_ \quad \ldots \quad \text{T}_{a_7} \quad \_
\]

There are 8 gaps in total: one before the first selected task, six between the selected tasks, and one after the last selected task.

### Step 3: Distribute the Gaps

To ensure no two selected tasks are adjacent, we must place at least one unselected task in each of the 6 gaps between the 7 selected tasks. Once one task is placed in each of the 6 gaps, there are \(8 - 6 = 2\) unselected tasks left to be distributed among the 8 gaps (including the ones at the ends).

### Step 4: Calculate the Number of Ways

The problem now reduces to determining the number of ways to distribute 2 remaining unselected tasks among the 8 available gaps. This is a classic "stars and bars" problem, where we have to place 2 identical stars (tasks) in 8 bins (gaps).

The number of ways to do this is given by the combination:

\[
\text{Number of ways} = \binom{8 + 2 - 1}{2} = \binom{9}{2}
\]

Calculating:

\[
\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36
\]

### Step 5: Conclusion

The total number of ways to select 7 tasks from the sequence of 15 tasks such that no two selected tasks are adjacent is **36**.

Would you like further explanation on any step, or do you have any questions?

---

Here are 5 related questions:
1. How would the number of ways change if the sequence had 20 tasks instead of 15?
2. What happens if you want to select 8 tasks with no adjacent selections?
3. How can the stars and bars theorem be applied to different combinatorial problems?
4. How would the solution be affected if the tasks were not distinct?
5. What if we required that no three selected tasks be adjacent?

**Tip:** In combinatorial problems involving constraints, breaking down the problem into manageable subproblems (like distributing gaps here) is often key to finding the correct solution.

Explore how to select 7 tasks from a sequence of 15 different tasks such that no two selected tasks are adjacent. Learn about combinatorial methods and the application of the Stars and Bars theorem in solving this problem. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation