Question

question 5 (1 point) how many ways can 7 friends stand in line for a movie if dima refuses to stand next to lara? 10,080 ways 40,320 ways 5,040 ways 1,440 ways 3,600 ways

Explanation:

Step1: Find total permutations of 7 friends

The total number of ways 7 friends can stand in a line is the permutation of 7 elements, which is \(7!\).
\(7! = 7\times6\times5\times4\times3\times2\times1 = 5040\)

Step2: Find number of permutations where Dima and Lara are together

Treat Dima and Lara as a single entity (since they must be next to each other). So now we have 6 entities to arrange (the Dima - Lara pair and the other 5 friends). The number of ways to arrange these 6 entities is \(6!\). But Dima and Lara can be in 2 orders within their pair (Dima - Lara or Lara - Dima), so we multiply by 2.
\(6! \times 2 = (6\times5\times4\times3\times2\times1)\times2 = 720\times2 = 1440\)

Step3: Subtract the unwanted permutations from total

The number of ways where Dima does not stand next to Lara is total permutations minus permutations where they are together.
\(7! - 6! \times 2 = 5040 - 1440 = 3600\)

Answer:

3,600 ways