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# Face Order

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:

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).

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.

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Age 14 to 16

ShortChallenge Level

- Problem
- 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:

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).

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.

You can find more short problems, arranged by curriculum topic, in our short problems collection.

A man went to Monte Carlo to try and make his fortune. Is his strategy a winning one?

Two bags contain different numbers of red and blue marbles. A marble is removed from one of the bags. The marble is blue. What is the probability that it was removed from bag A?

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?