Which of the following is the inverse z-transform ofX(z)

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Which of the following is the inverse z-transform of
X(z) = z |z| < 2
(z – 2) (z – 3)

1. [2ⁿ – 3ⁿ] u(– n – 1)
2. [3ⁿ – 2ⁿ] u(– n – 1)
3. [2ⁿ – 3ⁿ] u(n + 1)
4. [2ⁿ – 3ⁿ] u(n)

Correct Option: A

Given that

X(z) =	z	\|z\| < 2
	(z – 2) (z – 3)

or	X(z)	=	A	+	A
	z		z – 2		z – 3

or	X(z)	=	-1	+	1
	z		z – 2		z – 3

or X(z) = –		z		+		z
		z – 2				z – 3

or X(z) = –		1		+		1
		1 – 2z^{– 1}				1 – 3z^{– 1}

Now, since in the region |z| < 2 both the poles are exterior i.e., anti-causal and hence inverse z-transform.
x(n) = [– (– 2ⁿ) + (– 3ⁿ)] u(– n – 1)
or x(n) = (2ⁿ – 3ⁿ) u(– n – 1)
Hence, alternative (A) is the correct choice.