Kwak asked $120$ people to guess a number which was the permutation of $12345$ he was thinking about. Everyone tries to guess the correct permutation. $10$ people guess a permutation and are different from what Kwak thought in $5$ places (for example if the number Kwak thinks is $54321$, then someone guesses $12435$, then that person is wrong in $5$ places). $45$ people guessed a permutation and it was different from what Kwak thought in $4$ places. $45$ people guessed and differed in $3$ places. $15$ people guessed and differed in $2$ places. M is a number that indicates the number of people who managed to guess the number thought by Kwak correctly (there is no wrong place), and N is a number that indicates the number of people who guess a permutation and is different from what was thought by Kwak in $1$ place. What is the M-N value? Choices:
A. $5$
B. $4$
C. $3$
D. $2$
E. $1$
Next question: How many possible arrangements of permutation are different and there are no correct digits of placement (wrong in $5$ places)
We know that the number of permutations of $12345$ is $5! = 120$ numbers. Also we know $10 + 45 + 45 + 15 = 115$ are wrong answers. So, there are 5 answers left, i.e. 1 answer is the right answer and 4 the rest are wrong answers. We obtained $M = 1$ and $N = 4$, so $M-N =-3$.
But this is not the answer choice .. Somebody please help me..
No permutation differs in one place, because if four numbers are in the right places the fifth must be as well. Thus $N=0, M=5$ and $M-N=5$
You have confused the fact that $115$ people have wrong answers with the number of wrong answers that exist. We were not given that the various people chose different answers, so the number of answers is not relevant.