Fatorial #7

Algebra Level pending

Given the equation |3A| - $log 10^{27}$ + y! = $\frac{y!}{x! (y - x)!}$ ,

where A is and indentity matrix of order 3x3,

|3A| is the determinant of the matrix 3A,

(log 5 = 0.7), (log 3 = 0.45), (log 2 = 0.3) and (x! = x (x - 1)!).

First, find the two possible values of x (m and n), such as (y = 2x).

Then, consider that (m > n), (m = a + b), (n = a - b), (3m + 2n = c)

and find the value of (4a + 2b + c²).

The answer is 12.

1 solution

Tom Engelsman
Dec 21, 2020

The original equation above simplifies into:

$1 = \frac{1}{x!(y-x)!} \Rightarrow x!(y-x)! = 1$

which is satisfied by the points $(x,y) = (0,0); (0,1); (1,1); (1,2) \Rightarrow m = 1, n = 0$ .

If $m=a+b, n = a-b \Rightarrow 1 = a+b, 0 = a-b \Rightarrow a = b = \frac{1}{2}$ .

Also, $c = 3m+2n = 3(1)+2(0) = 3$ , which gives the final value:

$4a+2b+c^2 = 4(1/2)+2(1/2)+3^2 = 3+9 = \boxed{12}.$

thank u for ur solution! :D I just want to add a comment about the condition y=2x, which implies that the equation is satisfied only by (x,y)=(0,0); (1,2)

Victor Porto - 5 months, 3 weeks ago

Obrigado, Victor! I figured the condition m > n was sufficient for the values of x, but a good problem nonetheless. Happy Holidays & to more solutions in 2021 :)

tom engelsman - 5 months, 3 weeks ago

Fatorial #7

The answer is 12.

1 solution

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