Answer these questions based on the following information.
In an amphitheater there are 10 distinguishable seats placed in a row. A family of five wants to sit together, while rest of the attendees can sit anywhere without any restriction. In how many ways can 5 seats be selected so that this family can sit together?

Options

A. 4
B. 5
C. 6
D. 10

Answer

To solve the problem of how many ways a family of five can sit together in 10 distinguishable seats arranged in a row, we can follow these steps:

### Step 1: Understand the concept of seating together
When the family of five wants to sit together, we can treat them as a single unit or "block." This simplifies our problem because instead of arranging 5 individual family members, we can think of arranging 1 block of the family and the remaining seats.

### Step 2: Determine the number of blocks
Since the family of five is treated as one block, we now have:
- 1 block (the family)
- 5 remaining seats (since there are 10 seats in total)

This means we have a total of 6 blocks to arrange: 1 family block + 5 individual seats.

### Step 3: Calculate the number of ways to position the block
The family block can occupy any set of 5 consecutive seats. In a row of 10 seats, the family block can start in seat 1 and end in seat 5, or start in seat 2 and end in seat 6, and so on, until it starts in seat 6 and ends in seat 10.

To find the number of valid starting positions for the family block, we can use the formula:
\[ \text{Number of ways} = n - k + 1 \]
where \( n \) is the total number of seats (10) and \( k \) is the number of seats the family occupies (5).

### Step 4: Apply the formula
Substituting the values into the formula:
\[ \text{Number of ways} = 10 - 5 + 1 = 6 \]

### Step 5: Conclusion
Thus, there are 6 different ways for the family of five to sit together in the 10 seats.

### Final Answer
The number of ways the family can sit together is **6**.

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