Given '$$y$$' is a natural number, how many values of "$$y$$" are possible for $$(y^2 - 5y + 5)^{(y - 8)(y + 6)} = 1$$?

Expression : $$(y^2 - 5y + 5)^{(y - 8)(y + 6)} = 1$$

=> $$log(y^2 - 5y + 5)^{(y - 8)(y + 6)} = log(1)$$

=> $$(y-8)(y+6)log(y^2-5y+y)=0$$

Thus, either $$(y-8)=0$$ or $$(y+6)=0$$ or $$log(y^2-5y+5)=0$$

=> $$y=8,-6$$ ---------------(i)

or $$log(y^2-5y+5)=log(1)$$

=> $$y^2-5y+4=0$$

=> $$(y-4)(y-1)-0$$

=> $$y=4,1$$ -----------(ii)

Also, $$y^2-5y+5$$ can be equal to -1, and the power will be even; thus, it can be equal to 1.

$$y^2-5y+5$$ = -1

$$y^2-5y+6$$ = 0

y = 2 or 3

When y = 2, power will be(y-8)(y+6) = -48 even power, hence = 1

When y = 3, power will be 45 odd power, hence = -1

Thus, y = 2 also satisfies the condition.

From equations (i) and (ii), Possible values of $$y$$ (natural number) = 4

=> Ans - (A)