Derangements of the letters of the word PURPLE

Asked Jun 21 '21 at 15:34

Active Jun 25 '21 at 09:31

Viewed 71 times

In a Youtube video on the channel @blackpenredpen, the number of derangements of the word "PURPLE" is calculated as $$\frac{!6 \,- \,!5\, -\, !5\, - \,!4}{2!}$$ I don't understand two things:

Why is $!4$ being subtracted rather than being added, according to the Inclusion-Exclusion principle?
Why is the answer being divided by $2!$ despite the fact that the two Ps are accounted for already?

edited Jun 25 '21 at 09:31

Martin Sleziak

53,687

asked Jun 21 '21 at 15:34

Tuhin Mukherjee

1

See Number of Derangements of the word BOTTLE and maybe linked posts for more examples. – Vepir Jun 21 '21 at 15:39
5

I don't see why people are voting to close this question as "Missing context or other details". The OP mentioned the source (@blackpenredpen) which is a educational math channel, indicating motivation as self-learning. The OP also mentions they know of (tired) Inclusion-Exclusion, but it does not match to the given solution. I can agree that the formatting can (should) be better (@Tuhin Mukherjee, you should use latex to format equations, etc.), but also note that OP is a new contributor. – Vepir Jun 21 '21 at 15:49
Including a link to the video would be helpful. – N. F. Taussig Jun 21 '21 at 16:13
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I've tracked down the source video, and the pinned comment explains that the numerator should be $!6-(!5+!4)-(!5+!4)+!4$. @TuhinMukherjee does this clarify your first question? – Vepir Jun 21 '21 at 16:25
@Vepir why is it !5 + !4? – Tuhin Mukherjee Jun 21 '21 at 16:51

Derangements of the letters of the word PURPLE

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