Factorial Facts

[mmatlas]

1.Restate problem in your own words
2.Solve the problem.All math work needs to be showing
3Write a clear, complete,well organized esplanationthat tells how the problem was solved
4Include a chart or diagram or table or picture that helps explain the processs
5. write a conclusion

Factorial Fact

Y=(X-1)!
______
(X-3)!

What is the first X value which gives a y value over 10? over 100? over 1000 and over 10,000[/url][/u]

 

[arthur ohlsten]

y=[x-1]!/[x-3]!
but [x-1]!=[x-1][x-2][x-3]! or

y=[x-1][x-2]
y=x^2-3x+2
===============================================
x^2-3x+2>10
x^2-3x-8>0

let us first look at x^2-3x-8=0 and then when is it greater than 0

when does x^2-3x-8=0?
x=3+/-[9+32]^1/2 all over 2
x=1.5 +/-1/2 sqrt41

when is this greater than o?
-oo<x<1.5-.5sqrt41 and 1.5+.5 sqrt41<x<oo

you should now be able to answer the problem

Arthur

 

[soroban]

Hello, mmatlas!

\(\displaystyle \L y \;=\;\frac{(x-1)!}{(x-3)!}\)

What is the first \(\displaystyle x\) which gives a \(\displaystyle y\) over 10?
over 100? .over 1000? .over 10,000?

We have: \(\displaystyle \L\:y \;=\;\frac{(x-1)(x-2)(x-3)(x-4) \cdots3\cdot2\cdot1}{(x-3)(x-4)(x-5)\cdots3\cdot2\cdot1}\)

Reduce:\(\displaystyle \L\:y \;=\;\frac{(x-1)(x-2)(\sout{x-3})(\sout{x-4}) \cdots\not3\cdot\not2\cdot\not1}{\sout{(x-3)}\sout{(x-4)}\sout{(x-5)}\cdots\not3\cdot\not2\cdot\not1} \;=\;(x-1)(x-2)\)


We want \(\displaystyle (x-1)(x-2)\) to be greater than some constant, \(\displaystyle 10^n\)

We have: \(\displaystyle \x\,-\,1)(x\,-\,2) \:\>\: 10^n\;\;\Rightarrow\;\;x^2\,-\,3x\;+\,(2\,-\,10^n)\:>\:0\)

This is an up-opening parabola.
. . It is positive to the right of its greater x-intercept.

Quadratic Formula: \(\displaystyle \:x \;=\;\frac{-(-3)\,\pm\,\sqrt{(-3)^2\,-\,4(1)(2-10^n)}}{2(1)} \;=\;\frac{3\,\pm\,\sqrt{1\,+\,4\cdot10^n}}{2}\)


To be greater than \(\displaystyle 10^n:\;\;x\:>\:\frac{3\,+\,\sqrt{1\,+\,4\cdot10^n}}{2}\)


\(\displaystyle \begin{array}{ccccccc} > & | & \;n\; & | & \frac{3+\sqrt{1+4\cdot10^n}}{2} & | & \text{first }x \\ \hline
10^1 &|& 1 &|& \frac{3+\sqrt{41}}{2} \:=\:4.7015 &|& x\,=\,5 \\ \hline
10^2 &|& 2 &|& \frac{3+\sqrt{401}}{2}\:=\:11.5125 &|& x\,=\,12 \\ \hline
10^3 &|& 3 &|& \frac{3+\sqrt{4001}}{2}\:=\:33.1268 &|& x\,=\,34 \\ \hline
10^4 &|& 4 &|& \;\frac{3+\sqrt{40,001}}{2} \:=\:101.5013\; &|& x\,=\,102 \\ \hline\end{array}\)