Question

How many ways can you make a sandwich by choosing 4 out of 10 ingredients?

Answer 1

`10xx9xx8xx7= 5,040`

Explanation:

While there is a formula to use, it is quite easy to think through the question as follows:

When we choose the first ingredient, there are 10 choices.
When we now choose the second ingredient there are 9 choices.
For the third, there are 8.
For the fourth there are 7.

Each ingredient can be combined with the others, giving:
`10xx9xx8xx7= 5,040`

That's a lot of choices!

Answer 2

If the order does not matter 210
That is: cheese and tomato is the same as tomato and cheese
,..............................................................................................
If the order does matter 5040

Explanation:

`color(blue)("Explaining 'Factorial' by demonstration")`
`2"!" = 2xx1 = 2`
`3"!"=3xx2xx1 = 6`
`4"!"=4xx3xx2xx1=24`

`(4!)/(3!) = (4xx3xx2xx1)/(3xx2xx1) = (4xxcancel(3!))/(cancel(3!))=4`

Sometimes you will see factorial written as `->4! " is the same as "ul(4)|`

So `(4xxcancel(ul(3|)))/(cancel(ul(3|)))=4`

,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

Depends on how you wish to consider the paring up.

`color(brown)("If you consider type a+b as being the same as type b+a then this is")` `color(brown)("called a "color(magenta)("COMBINATION. "))`

This can be written as `color(white)()^10C_4` or alternatively `((10),(4))`. I much prefer `color(white)()^10C_4` as there is no doubt what is meant.

`color(white)()^10C_4 = (10!)/((10-4)!4!) = (10xx9xx8xx7xxcancel(6!))/(cancel(6!)4xx3xx2)`

`color(green)(color(white)()^10C_4 = 210)`
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("If you consider type a+b as NOT being the same as type b+a then this is")` `color(brown)("called a "color(magenta)("PERMUTATION. "))`

This can be written as `color(white)()^10P_4`

`color(white)()^10P_4 = (10!)/((10-4)!) = (10xx9xx8xx7xxcancel(6!))/(cancel(6!)) = 5040`

`color(green)(color(white)()^10P_4=5040)`