0

In how many ways can one put 5 letters into 5 envelopes correctly if 1 letter alone should go into the correct Envelope?

miracle173
  • 11,049
Anand
  • 21
  • 1
    Can you post a picture of the question if it's from a book or something? I can't understand the question. – mihir Nov 19 '15 at 13:55
  • I am Sorry... See if u can understand now – Anand Nov 19 '15 at 13:57
  • @mihir: a picture of the letters and the envelopes? – miracle173 Nov 19 '15 at 14:03
  • Is your question that you have five letters: $a,b,c,d,e$, and five envelopes $A,B,C,D,E$, and you send one letter to each of the envelopes in some manner, say $\begin{pmatrix} A&B&C&D&E\a&c&d&e&b\end{pmatrix}$ in such a way that only one envelope received the correct letter? – JMoravitz Nov 19 '15 at 14:04
  • @miracle173, what do you mean? Since the original post wasn't worded properly, I couldn't understand the question, and that's why I asked him to post a picture of the question if it's from a book – mihir Nov 19 '15 at 14:07
  • @mihir: A picture of a text is not a good idea. This is not wanted here. He can type the original text to this post. If he needs assistance in formatting the text he will get help here. If one selects the context menu (right mouse button) 'Show -Math -> TeX Commands' of a TeX graphic, one can see the TeX command that generates this formula. TeX commands must be preceded and terminated by a dollar character (or two dollar characters is it should be positioned in a separate line) – miracle173 Nov 19 '15 at 14:55

3 Answers3

1

Its a typical example of dearrangement the general formula is $D_n=n![1-\frac{1}{1!}+..-..+\frac{(-1^{n})}{n!}]$ so here dearrangements are 4 so plug in the value as $n=4$ and you get answer as $9$ . Hope it helps you. But from next time please share your effort!

Archis Welankar
  • 15,910
0

Since I don't want to give out the answer to anyone who may be trying to work on this problem, think of the analogous problem: How any ways can you place 4 letters into 4 envelopes so that none are in their correct envelope?

Yunus Syed
  • 776
0

From what I understand, choose the letter to be posted correctly in 5 ways.

Then derange the other 4. D(n) can be derived by inclusion exclusion principle (it is a simple exercise) as $$ D(4) = 9.$$

Therefore, Number of ways = 5*9=45

Tushant Mittal
  • 847
  • Could u use normal permuation and explain... – Anand Nov 19 '15 at 14:05
  • To find D(n) use exclusion -inclusion as I have mentioned or the link to a proof is here, https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle#Examples and for a recursive proof http://math.stackexchange.com/questions/1056610/combinatorial-argument-for-recursive-formula – Tushant Mittal Nov 19 '15 at 14:13