To solve the problem of seating eight people in a row where two specific people must sit next to each other, we can treat the two people who want to sit together as a single entity or block.
Let’s denote the two people who want to sit together as A and B. By treating them as one block, we can consider this block as a single unit. This means instead of arranging 8 individual people, we are now arranging 7 units: the block (AB) and the other 6 individuals (C, D, E, F, G, H).
The 7 units can be arranged in 7! (factorial of 7) different ways. Calculating that gives:
- 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
However, within the block (AB), A and B can also switch places, meaning we have 2! arrangements for A and B. This gives:
- 2! = 2 × 1 = 2
Now, to find the total number of arrangements where A and B are sitting together, we multiply the two results:
- Total arrangements = 7! × 2! = 5040 × 2 = 10080
Therefore, there are 10,080 ways to seat the eight people such that the two specific individuals sit next to each other.