Friends Sitting Together

Solution & Explanation

To solve the problem of finding the probability that all 10 friends sit consecutively in a row of 20 seats, we need to consider both the total number of possible seating arrangements and the number of favorable arrangements where the friends sit together.

Step-by-Step Breakdown:

1. Total Possible Arrangements:

   • We begin by calculating the total number of ways to choose 10 seats out of 20 for the friends.

   • This is given by the combination formula $\binom{20}{10}$, which calculates the number of ways to choose $k$ items from $n$ items without regard to order:

     $\binom{20}{10} = \frac{20!}{10! \cdot 10!}$

   • Using a calculator, this evaluates to $184,756$ ways.

2. Favorable Arrangements (Friends Sitting Together):

   • To ensure all 10 friends sit next to each other, consider them as a single block or unit.
   • This block can be placed in different positions along the row.
   • There are 11 possible starting positions for this block of 10 seats:
     • Seats 1 to 10
     • Seats 2 to 11
     • Seats 3 to 12
     • ...
     • Seats 11 to 20
   • Thus, there are 11 ways to arrange the block of friends consecutively.

3. Calculate the Probability:

   • The probability is the ratio of the number of favorable arrangements to the total number of possible arrangements:

     $\text{Probability} = \frac{\text{Number of favorable arrangements}}{\text{Total number of possible arrangements}}$

   • Substituting the values:

     $\text{Probability} = \frac{11}{184,756}$

   • This simplifies to approximately $0.0000596$, or about 0.00596%.

Explanation of Steps:

• Why 11 Positions?

  • The friends, treated as a single block, can start at position 1 and go up to position 11, ensuring they occupy 10 consecutive seats each time.

• Why Use Combinations?

  • We use combinations because the order of choosing which seats the friends occupy doesn't matter — only which seats are occupied.

• Simplification:

  • The $10!$ terms cancel out when considering the arrangement of friends within the block because we're only interested in their collective positioning.

Conclusion:

The probability of all 10 friends sitting consecutively in a row of 20 seats is extremely low, at approximately 0.00596%. This highlights the rarity of such an arrangement occurring purely by chance.