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There are $7$ switches on a switchboard, some of which are on and some of which are off. In one move, you pick any $2$ switches and toggle each of them-if the switch you pick is currently off, you turn it on, if it is on, you turn it off. Your aim is to execute a  sequence of moves and turn all $7$ switches on. For which of the following initial configurations is this not possible? Each configuration lists the initial positions of  the $7$ switches in sequence, from switch $1$ to switch $7.$

  1. (off,on,off,on,off,off,on)
  2. (off,on,on,on,on,on,off)
  3. (on,off,on,on,on,on,on)
  4. (off,off,off,off,off,on,off)
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  $S_1$ $S_2$ $S_3$ $S_4$ $S_5$ $S_6$ $S_7$ $Toggled Pairs$ $Are\ all \ switches \ ON \ after\ toggle?$
$a.$ $off$ $on$ $off$ $on$ $off$ $off$ $on$

$(S_1,S_3) $ and $(S_5,S_6)$

$YES$
$b.$ $off$ $on$ $on$ $on$ $on$ $on$ $off$ $(S_1,S_7) $ $YES$
$c.$ $on$ $off$ $on$ $on$ $on$ $on$ $on$   $NO$
$d.$ $off$ $off$ $off$ $off$ $off$ $on$ $off$ $(S_1,S_2) $ and $(S_3,S_4)$ and $(S_5,S_7)$ $YES$

In the given options we just select a pair of switches each time which are $off$ and turn them $on$.

For making a pair the number of  switches that are $off$ should be even.

Only option $C.$ violates this hence Option $C.$ is the correct answer.

Answer:

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