I often solve math questions because I like it (This may sound crazy, I know :)). Today I came across an interesting permutation & combination question.
The question is as follows:
6 people (named A, B, C, D, E, F) are in a line in a supermarket. In
how many possible arrangements is C between A and B?Options: A. 30 B. 60 C. 120 D. 240 (<- Correct answer) E. 360
The previous question was like: "In how many possible arrangements C is in front of A". My answer was: 6!/2 which is 360. Because in one half, C is in front of A, and in the other half A is in front of C. (No calculation required, common sense & interpretation & mathematical intuition is sufficient to find the correct answer I believe.)
But this one really puzzled me. I will be grateful if you provide a clear explanation.
Is there any way to solve this question without writing down all combinations? Any practical method? Because this question is from a math book which prepares students for the university entrance exam in my country in which every question should not take more than 30 sec to be solved.
Let me know if the questions is unclear (because I translated it, probably in a bad way)
Many many thanks in advance.
Best Answer
All you care about is the order of $A,B,C$. There are $3!=6$ orders of them,of which $2$ have $C$ in the middle. So $\frac 26=\frac 13$ of the arrangements have $C$ in the middle. $\frac {6!}3=240$