The expressions are illustrations of permutations
The first expression is given as: 6!
Using factorial formula, we have:
[tex]6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1[/tex]
Multiply the factors
[tex]6! = 720[/tex]
This means that, there are 720 ways to line for a photo
The next expression is given as:
[tex]\frac{6!}{(6 - 2)!}[/tex]
Rewrite as:
[tex]\frac{6!}{(6 - 2)!} =\frac{6!}{4!}[/tex]
Expand
[tex]\frac{6!}{(6 - 2)!} =\frac{6 \times 5 \times 4!}{4!}[/tex]
Divide the common factors
[tex]\frac{6!}{(6 - 2)!} =6 \times 5[/tex]
Multiply
[tex]\frac{6!}{(6 - 2)!} = 30[/tex]
This means that, there are 30 ways to give responsibilities to two of the six people
The next expression is given as:
[tex]\frac{6!}{(6 - 3)!3!}[/tex]
Rewrite as:
[tex]\frac{6!}{(6 - 3)!3!} = \frac{6!}{3!3!}[/tex]
Expand
[tex]\frac{6!}{(6 - 3)!3!} = \frac{6 \times 5 \times 4 \times 3!}{3!\times 3 \times 2 \times 1}[/tex]
Divide the common factors
[tex]\frac{6!}{(6 - 3)!3!} = \frac{6 \times 5 \times 4}{ 3 \times 2 \times 1}[/tex]
[tex]\frac{6!}{(6 - 3)!3!} = \frac{120}{6}[/tex]
Divide
[tex]\frac{6!}{(6 - 3)!3!} = 20[/tex]
This means that, there are 20 ways to give three people can be asked to sit in front
Read more about permutations at:
https://brainly.com/question/8119212
