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The recurrence relation capturing the optimal execution time of the $Towers \ of \ Hanoi$ problem with $n$ discs is

1. $T(n) = 2T(n − 2) + 2$
2. $T (n) = 2T(n − 1) + n$
3. $T (n) = 2T(n/2) + 1$
4. $T (n) = 2T(n − 1) + 1$
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There are three towers named as Left, Middle, Right.
Tower Left contains 3 discs in decreasing size from bottom to top.
We have to move these 3 discs from Left tower to Right tower using Middle tower.
step 1: Move 2 discs from Left tower to Middle tower using Right tower. $n-1$ discs
Step 2: Move the remaining disc from Left tower to Right tower. $1-movement$
Step 3: Move 2 discs from Middle tower to Right tower using Left tower. $n-1$ discs

Recurrence relation:
$T(n) = T(n-1) + T(n-1) + 1$
$T(n) = 2T(n-1) + 1$

Recurrence relation for Towers of Hanoi is

$T(1) = 1$

$T(n) = 2 T( n-1 ) +1$

So Answer should be (D)

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