Factorial says 'Happy New Year'

Number Theory Level 2

Find approximate value of $\dfrac{2020!+2017!}{2019!+2018!}$ .

Notation: $!$ denotes the factorial notion . For example: $8! = 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8$ .

2018 2020 2019 2017

2 solutions

Chew-Seong Cheong
Jan 2, 2019

$\begin{aligned} \frac {2020!+2017!}{2019!+2018!} & = \frac {2017!(2020 \cdot 2019 \cdot 2018 +1)}{2018!(2019+1)} \\ & = \frac {2020 \cdot 2019 \cdot 2018 +1}{2018\cdot 2020} \\ & = 2019 + \frac 1{2018 \cdot 2020} \\ & \approx \boxed{2019} \end{aligned}$

Jahangir Hossain
Jan 1, 2019

given,

$\space \space \space \space \frac{2020!+2017!}{2019!+2018!}$

$=\frac{(2020 \times 2019 \times 2018 \times 2017!)+2017!}{(2019 \times 2018 \times 2017!)+(2018 \times 2017!)} \space \space [ \space \space using \space \space n!=n \times (n-1)! \space \space]$

$=\frac{2017!(2020 \times 2019 \times 2018 +1)}{2017!(2019 \times 2018+2018)}$

$=\frac{(2020 \times 2019 \times 2018) +1}{(2019 \times 2018)+2018}$

$=\frac{8230170841}{4076360}$

$=\boxed{2019}$

How did you cancel out in the last step?

Henry U - 2 years, 5 months ago

After cutting off 2017!, I used a calculator to get the value 😫

Jahangir Hossain - 2 years, 5 months ago

You can factor out 2018 in the denominator and then cancel out

$\begin{aligned} & \frac {2020\cdot 2019\cdot 2018+1}{2019\cdot 2018+2018} \\ & = \frac {2020\cdot 2019\cdot 2018+1}{2018\cdot 2020} \\ & =\frac{2020\cdot 2019\cdot 2018}{2018\cdot 1020}+\frac{1}{2018\cdot 2020} \\ & = 2019+\frac{1}{2018\cdot 2020} \end{aligned}$

Henry U - 2 years, 5 months ago

thanks sir.

Jahangir Hossain - 2 years, 5 months ago

Factorial says 'Happy New Year'

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