Four letters are written and the respective addresses are also written on four envelopes. In how many ways can the letters be inserted into the envelopes so that no letter is in the right envelope?
Answers
Answer:
9 ways
Explanation:
There are 4 address and 4 envelopes.
The number of ways 4 letter can be inserted into the 4 envelopes = 4! = 4 x 3 x 2 x 1 = 24 ways
Now, we need to find the number of ways in which at least 1 letter is in the correct envelope.
At least 1 letter in the correct envelope has the following possibilities:
1) 1 letter in the correct envelope
2) 2 letters in correct envelopes
3) 3 letters in correct envelopes
4) 4 letters in correct envelopes
There is a point to be noted here.
3 letters in their correct envelope keep one letter and one envelope free and eventually, the 4th letter will be in the correct 4th envelope.
So, point 3 can be omitted as point 3 and 4 means the same.
Let the letters be A, B, C, D for envelopes E1, E2, E3, E4.
1 letter in the correct envelope:
Below are the possible ways
E1 E2 E3 E4
A D B C
A C D B
C B D A
D B A C
B D C A
D A C B
B C A D
C A B D
= 8 WAYS
2 letters in correct envelopes:
Below are the possible ways
E1 E2 E3 E4
A B D C
A D C B
A C B D
D B C A
C B A D
B A C D
= 6 WAYS
4 letters in correct envelopes:
E1 E2 E3 E4
A B C D
= 1 WAY
Therefore, the number of ways all the letters are in the wrong envelopes = Total number of ways the letters can be placed into the envelopes — The number of ways at least one letter is correctly placed
= 24 - (8 + 6 + 1)
= 24 — 15
= 9 ways.