Answer:
[tex]Pr = 6.67\%[/tex]
Step-by-step explanation:
Given
[tex]n = 10[/tex] i.e. 10 people
Required
Probability of being next to each other
First, we calculate the total possible arrangements (without any restriction)
[tex]Total = n![/tex]
[tex]Total = 10![/tex]
If the three are to be next to each other,
First, we arrange the three
[tex]r_1 = 3![/tex]
Now, the 3 will be seen as 1; so, we have a total of 8 people i.e. (1 + 7 others)
The arrangement is:
[tex]r_2 =8![/tex]
So, the total arrangement, when they have to be next to one another is:
[tex]r = r_1 * r_2[/tex]
[tex]r = 3! * 8![/tex]
The probability is:
[tex]Pr = \frac{r}{n}[/tex]
[tex]Pr = \frac{3! * 8!}{10!}[/tex]
Expand
[tex]Pr = \frac{3*2*1 * 8!}{10*9*8!}[/tex]
[tex]Pr = \frac{3*2*1}{10*9}[/tex]
[tex]Pr = \frac{6}{90}[/tex]
[tex]Pr = 0.0667[/tex]
Express as percentage
[tex]Pr = 0.0667 * 100\%[/tex]
[tex]Pr = 6.67\%[/tex]
