desaichinmay22 wrote:
5 letters have to be put into their 5 respective envelopes. If the letters are inserted randomly into the envelopes, what is the probability that all the letters go into their respective envelopes?
A)1/30
B)1/60
C)1/120
D)1/240
E)3/5
let 5 envelopes be E1,E2,E3,E4,E5. now first letter can go to any of 5 envelopes in 5 ways. second letter can go to any of the 4 remaining envelopes in 4 ways. third letter can go to remaining 3 envelopes in 3 ways. fourth letter can go to remaining 2 envelopes in 2 ways. and finally last envelopes has only 1 ways.
so when no restrictions are applied. these 5 letters can arrange themselves in 5 envelopes in 5.4.3.2.1=120 ways
(we can also think of this as arrangements of 5 students in 5 vacant spaces.)
now, obviously there is only one correct envelope designated for each letter. thus the number of ways in which all letters will be correctly posted=1
thus required probability = 1/120