In how many ways can a photographer at a wedding arrange 6 people in a row

This is really a mathematical question, not a photography question. You have the choice of Permutation - Wikipedia and Combination - Wikipedia. A combination just counts the people, while a permutation also includes the sequence in which they stand. Clearly combinations are easier. The formula is n!/(k!*(n-k)!). ! is the Factorial - Wikipedia function, the number multiplied by all smaller numbers. Let's take the individual cases:

1. Bride must be in the picture. That means 5 others out of 9. So if you have 9 people and you want to choose 5, the number is 9!/(5!*(9-5)!), or 362880 / (120 * 24), or 126. This is often written 9C5

2. Both bride and groom must be in the picture. In this case you need 4 others out of 8, so the combination formula is 8C4, or 70.

3. Either bride or the groom must be in the picture, but not both. That means 5 other people out of 8, or 8C5, 56.

I’ll leave the permutations as an exercise for the reader. My (cheap) scientific pocket calculator has convenient functions on the keyboard. If you have a scientific calculator, yours might too.

This is the concept of a piecewise function. for your function, you have



the second halves tell you the interval in which the left equation is graphed in. they are defined as such. they are there because whoever made the notes/question/whatever the picture is literally defined the function to be like that. the graph ends up looking like that.

means between  (inclusive)  and  (inclusive). 

means between  (exclusive)  and  (inclusive). 

for example, between -2 and 3 (inclusive), the function graph will have the equation , which is a line with y-intercept of 1. then, between 3 (exclusive) and 7 (inclusive), the function graph will have equation  , which is a parabola with vertex at (5,0).

as for why those two definitions of the function can mean the same thing:



and 



and have the same value at  , so it does not matter which piece gets the "equal to." notice how they changed which piece gets the  (less than or equal to).

if both pieces were to be defined at  , then the function would not be a function (because it would technically have one domain element mapping to two range elements). so we have to leave one piece that doesn't include  .


i have attached another example of a piecewise function.

note that piecewise functions are not just limited to straight lines and parabolas; they can be anything that you can graph.