3 Bob has been given a pile of five letters addressed to five different people. He has also been given a pile of five envelopes addressed to the same five people. Bob puts one letter in each envelope at random.

1. How many different ways are there to pair the letters with the envelopes?
2. Find the number of arrangements with exactly three letters in the correct envelopes.
3. (a) Show that there are two derangements of the three symbols A , B and C .
   (b) Hence find the number of arrangements with exactly two letters in the correct envelopes. Let \(\mathrm { D } _ { n }\) represent the number of derangements of \(n\) symbols.
4. Explain why \(\mathrm { D } _ { n } = ( n - 1 ) \times \left( \mathrm { D } _ { n - 1 } + \mathrm { D } _ { n - 2 } \right)\).
5. Find the number of ways in which all five letters are in the wrong envelopes.