Question

in how many ways can nine people be lined up for a photograph if clark and billy must not stand next to each other

Answer 1

Consider Clark and Billy to be bound together as one person. There are two ways to do this: Clark on the left and Billy to his right or Clark on the left and Billy to his left.

Answer 2

If the question were the following:

How many ways can nine people be lined up for a photograph if clark and billy MUST stand next to each other, then the answer would be 8! * 2.

Answer 3

With Clark and Billy counting as one person, there are 8 persons left. Those people can be placed in a linear formation in the following number of ways:

8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 or 8!

That comes from there being 8 choices for the first person in line, then 7 for the second and as the selections continue, 1 person remains for the 8th position.

Answer 4

Because Billy/Clark can be arranged bound together in 2 ways as described above, then the number of ways nine people be lined up for a photograph if clark and billy MUST stand next to each other, then the answer would be 8! * 2.
8! * 2 = 80 640.

Answer 5

Consider this question:

In how many ways can nine people be lined up for a photograph?

That would be 9! = 362 880 ways with Billy and Clark located in any one of the 9 positions including next to each other.

Answer 6

The number of ways for the 9 people to form a line - the number of ways the 9 people can form a line with Billy/Clark together = the number of ways the 9 people can form a line with Billy and Clark not standing together.

Answer 7

9! - 8!*2 = 362 880 - 80 640 = ?

Answer 8

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