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Parity announcements in a row

BrainteasersEasy

The warden gives each of $5$ prisoners a red or a blue hat, all $2^{5} = 32$ colour patterns equally possible. The prisoners stand in a line facing the same way and each sees every hat in front of them but not their own or those behind. Each prisoner announces a colour by this fixed rule: say red if the number of red hats in front of you is even, otherwise say blue. For how many of the $32$ patterns are exactly $3$ of the $5$ announcements correct?

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Hint

Fix the hats from the front of the line backwards. Given the hats in front of a prisoner, how many of the $2$ colours make his announcement correct?

Solution

Choose the hats from the front prisoner backwards. Once the hats in front of a prisoner are fixed, his announcement is fixed, so exactly $1$ of his $2$ possible colours makes it correct.
So each pattern gives exactly one pattern of right and wrong announcements, and every right-or-wrong pattern for the $5$ prisoners occurs exactly once.
The patterns with exactly $3$ correct announcements correspond to choosing which $3$ of the $5$ prisoners are right, which gives $(3 5)$ choices.
Hence the number of patterns is $10$ .

Answer: $10$

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