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

2 Answers
Aug 2, 2016

Answer:

#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!

Aug 2, 2016

Answer:

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)#