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I have an exercise that has factorial and as I know

$10!$

I must do it $10 \cdot 1 + 10 \cdot 2 \ldots$ etc and I don't know how I see this solution. He just written the results,or calculates with another way,faster way? I will use a computer to calculate it but i don't know how he fast did it. \begin{align*} P(n,r) & = \frac{n!}{(n-r)!}\\ & = \frac{10!}{(10 - 3)!}\\ & = 10 \times 9 \times 8\\ & = 720 \end{align*}

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  • $\begingroup$ Please read this MathJax tutorial, which explains how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig Jun 21 '18 at 22:36
  • $\begingroup$ you don't see?it doesn't mention it that it says 10 * 9 * 8 .As you can see we have 10! /7! that leads to this (10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1)/(1*2*3*4*5*6*7 ).How the 10*9*8 stays? $\endgroup$ – m.s Jun 21 '18 at 22:37
  • $\begingroup$ omg now i understand impossible xaxaxax $\endgroup$ – m.s Jun 21 '18 at 22:37
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    $\begingroup$ he divide 1 until 7 / 7 down and up remain 8 *9*10 . $\endgroup$ – m.s Jun 21 '18 at 22:38
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    $\begingroup$ How the $10⋅9⋅8$ stays? Because $7⋅6⋅5⋅4⋅3⋅2⋅1$ is present in the numerator and in the denominator and so they cancel out. $\endgroup$ – Weather Vane Jun 21 '18 at 22:48
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Notice that $$10!=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$$ while: $$7!=7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$$ hence it is clear that $$\frac{10!}{7!}=10\cdot9\cdot8=720$$

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