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

Aug 2, 2016

$10 \times 9 \times 8 \times 7 = 5 , 040$

#### 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:
$10 \times 9 \times 8 \times 7 = 5 , 040$

That's a lot of choices!

Aug 2, 2016

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

#### Explanation:

$\textcolor{b l u e}{\text{Explaining 'Factorial' by demonstration}}$
$2 \text{!} = 2 \times 1 = 2$
$3 \text{!} = 3 \times 2 \times 1 = 6$
$4 \text{!} = 4 \times 3 \times 2 \times 1 = 24$

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

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

So $\frac{4 \times \cancel{\underline{3 |}}}{\cancel{\underline{3 |}}} = 4$

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$\textcolor{b l u e}{\text{Answering the question}}$

Depends on how you wish to consider the paring up.

$\textcolor{b r o w n}{\text{If you consider type a+b as being the same as type b+a then this is}}$color(brown)("called a "color(magenta)("COMBINATION. "))

This can be written as ${\textcolor{w h i t e}{}}^{10} {C}_{4}$ or alternatively $\left(\begin{matrix}10 \\ 4\end{matrix}\right)$. I much prefer ${\textcolor{w h i t e}{}}^{10} {C}_{4}$ as there is no doubt what is meant.

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

$\textcolor{g r e e n}{{\textcolor{w h i t e}{}}^{10} {C}_{4} = 210}$
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$\textcolor{b r o w n}{\text{If you consider type a+b as NOT being the same as type b+a then this is}}$color(brown)("called a "color(magenta)("PERMUTATION. "))

This can be written as ${\textcolor{w h i t e}{}}^{10} {P}_{4}$

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

$\textcolor{g r e e n}{{\textcolor{w h i t e}{}}^{10} {P}_{4} = 5040}$