Question

question 2 (1 point) how many ways can 8 friends stand in line for a movie if dima wants to stand next to lara? 30,240 ways 5,040 ways 40,320 ways 10,080 ways 35,280 ways page 2 of 10

Explanation:

Step1: Treat Dima and Lara as a single unit

Since Dima wants to stand next to Lara, we can consider them as one combined entity. So now we have \(8 - 1=7\) units to arrange (the Dima - Lara unit and the other 6 friends). The number of ways to arrange \(n\) distinct units is \(n!\). So the number of ways to arrange these 7 units is \(7!\).

Step2: Account for the internal arrangement of Dima and Lara

Within the Dima - Lara unit, Dima and Lara can be arranged in 2 ways (Dima next to Lara or Lara next to Dima), which is \(2!\) (since \(2!=2\times1 = 2\)).

Step3: Calculate the total number of arrangements

To find the total number of ways, we multiply the number of ways to arrange the 7 units by the number of ways to arrange Dima and Lara within their unit. So the total number of arrangements is \(7!\times2!\).
We know that \(n!=n\times(n - 1)\times\cdots\times1\), so \(7!=7\times6\times5\times4\times3\times2\times1 = 5040\) and \(2!=2\times1=2\). Then \(7!\times2!=5040\times2 = 10080\).

Answer:

10,080 ways