Question

A safe has 5 locks, v, w, x, y and z ; all of which must be unlocked for the safe to open. The keys to the locks are distributed among five executives in the following manner :

Mr. P has keys for locks v and x.
Mr. Q has keys for locks v and y.
Mr. R has keys for locks w and x.
Mr. S has keys for locks x and z.
Mr. T has keys for locks v and z.

The minimum number of executives required to open the safe is _____

• A. 3

Answer 1

The correct option is A 3
The following table indicates the executives and the locks they can open.

Executives	Locks
	$v$	$w$	$x$	$y$	$z$
$P$	$\prod$		$\prod$
Q	$\prod$			$\prod$
R		$\prod$		$\prod$
S			$\prod$		$\prod$
T	$\prod$				$\prod$

From the truth table we can identify that key for locks w is only with Mr. R. So Mr. R is the essential executive, without whom the safe cannot be opened. If R is present, he can open lock y also. As seen from the table, the remaining locks v, x and z can be opened by P and S (or) P and T (or) Q and S (or) S and T. So the combinations of executives who can open the locks are RPS (or) RPT (or) RQS (or) RST. Hence, the Boolean expression is
F(P, Q, R, S, T) = RPS + RPT +RQS +RST
i.e., the minimum number of executives required to open the safe is 3.