Question

I'm sleepy. But I still don't understand why this doesn't work.

Answer 1

This is a problem on derangements.

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

Answer 2

if your sleepy get some coffee ok

Answer 3

^

Answer 4

or do want to go to sleep

Answer 5

Now let's start by placing card 1 into envelope 2.
After we're done with that easy step, we know that wherever we place our card 2, it's definitely not gonna go into envelope 2. There are exactly \(5\) places remaining for it to go.
Now let's say it goes into envelope \(x\). Then we know that card \(x\) can go to any of the four remaining places.

AND JUST AS I WAS TYPING, I REALISED THAT THIS DIDN'T WORK BECAUSE I DOUBLE-COUNTED.

Answer 6

PARTHFACE! GO SLEEP! YOU WORRY ME KID.

Answer 7

^

Answer 8

After, 1-2 you are choosing to place 5 card into 5 envelopes and you have 4 excluded combinations for same card into same env.
\(\color{#000000}{ \displaystyle 5~nCr~5~-4 }\)

could be wrong, but this is my first impression.
got to go...

Answer 9

ERR I just discarded what I was typing.
Anyway, the point is that we've got to get some inclusion and exclusion here.

Answer 10

this is eazy if think about it