face order
How many ways can these five faces be ordered?
Age
14 to 16
Challenge level
Problem
How many ways can the five faces below be ordered if the smiling face cannot be on either end?
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This problem is taken from the World Mathematics Championships
Student Solutions
Since the smiling face is the one with a restriction, we could put that one in first. Since it can't go at either end, there are 3 options for its space:
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Then moving onto the next face, whichever of the 3 options we chose for the smiling face, there are 4 options for this face (any of the 4 remaining spaces).
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So since there are 4 options corresponding to each of the 3 options for the smiling face, that gives a total of 4$\times$3 = 12 options for the first two faces.
Then there will be 3 spaces left in which to put the next face, then 2 spaces, and finally only 1 space left for the last face.
So altogether there are 12$\times$3$\times$2$\times$1 = 72 options to place all of the faces.