A) in how many ways can 6 people be lined up to get on a bus? (b) if 4 specific persons, among 6, insist on following each other, how many ways are possible? (c) if 2 specific persons, among 6, refuse to follow each other, how many ways are possible?

a) 720

b) 144

c) 480

Step-by-step explanation:

a) 6 people changing places in a line is permutation (because the a line has a order)

This way, 6 people in a line is 6! = 6.5.4.3.2.1 = 720

b) We can consider 4 people a block that will permut, so, instead of 6 people we have 3 [a block of 4 people and 2 more):

3! = 3.2.1 = 6

This group of 4 can also permut, so 4! = 4.3.2.1 = 24

Total = 3!.4! = 6.24=144

c) For this case, the easiest way is doing the complement. Calculate how many the 2 people are together and subtract from the total, so:

We can consider 2 people a block that will permut, so, instead of 6 people we have 5 [a block of 2 people and 4 more):

5! = 5.4.3.2.1 = 120

The 2 people from the block will permut as well:

2! = 2.1 = 2

Total: 5!.2! = 120.2 = 240.

For the case they are not following each other:

720 - 240 = 480

the ratio of balloons to tables is 5: 2 and the constant variation is 5.

the equation is y=5x