In Jane's house one room has 7 lights in a row. Each light can either be switched ON or OFF. Jane makes the following two statements: 1. More than 2 Lights are OFF 2. Any OFF Light must not have exactly one adjacent ON light. Given that 1 is False and 2 is TRUE, how many different unique arrangements are possible? 1. 5 2. 9 3. 12 4. 16

Statement 1 false means we can have 0 or 1 or 2 lights off = 7 or 6 or 5 lights on. I worked out 9 patterns (o on x off) 7 on = ooooooo 6 on combos noting no 1s allowed 0-6 = xoooooo 2-4 = ooxoooo 3-3 = oooxooo 5 on combos 0-0-5 = xxooooo 0-2-3 = xooxooo 0-3-2 = xoooxoo 0-5-0 = xooooox 2-0-3 = ooxxooo This assumes mirror images (eg. 4-2 vs 2-4, 3-0-2 vs 2-0-3) don't count. If included they add up to 15 but is not in the answers.

its 16 because the combinations go upto more than 12 lol. but i wouldnt waste ime on that question if its just 1 pointer,

I don't necessarily agree that statement one being FALSE means there are 0, 1, or 2 lights on. I'd interpret it as meaning more than two lights are on... how do you come to your conclusion, @A1? What am I interpreting incorrectly?

All questions are worth more than one point. Just to clarify, do you mean if there weren't multiple questions based on the same stimulus?

More than 2 lights on can make but does not guarantee the statement false. Like 4 on + 3 off, the latter makes it true.

So this is the way I saw it. From Rule 1, we say that 0,1 or 2 lights are off. 'Any OFF Light must not have exactly one adjacent ON light.' If we take this as xxxxxxx, we can already exclude the off lights from being on either end of the sequence as this would mean there is only 1 adjacent ON light if the on light was 2nd in the sequence. So you would be left with five possible places for the off lights. xxxxxxx x0xxxxx x0x0xxx x0xx0xx x0xxx0x xx0x0xx xx0xx0x xxx0x0x xx0xxxx xxx0xxx xxxx0xx xxxxx0x ....actually i just realised its a hec more than five Guess the answer was 12 C

Oops my interpretation above of the rule 2 was incorrect. Indeed it means an Off light must be between two On lights, and 12 is the answer.

Here's my attempt at it. So if you have two statements: 1. More than 2 Lights are OFF - False (essentially, you have the statement that there are 0, 1, or 2 lights off) 2. Any OFF Light must not have exactly one adjacent ON light for 0 lights off, there is 1 combination (all on) for 1 light off, there are exactly five combinations (the single off light being anywhere except either end) for 2 lights off, as per Rule 2, the off lights cannot be on either end and the off lights cannot be adjacent to each other. Basically this means in the middle 5 lights, you have (where 1 is an OFF light) lights 4 apart x 1 10001 lights 3 apart x 2 10010 and 01001 lights 2 apart x 3 10100 01010 00101 This gives the requisite 12.