# Thread: Letters with wrong address

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## Letters with wrong address

A secretary types 4 letters and then addresses the 4 corresponding envelopes. In how many ways can the secretary place the letters in the envelopes so that NO letter is placed in its correct envelope

A.8
B.9
C.10
D.12
E.15

2. Good post? |
I get 12.

Total = Correct + Incorrect.

4*4 = Correct + Incorrect.

Incorrect = 16-4 = 12.

3. Good post? |
None match the answer

Should be 23

Total ways of putting 4 letters in 4 envelopes (assuming an envelope can only hold one letter) = 4 * 3 * 2 *1 = 24

Total ways of putting all letters in their correct envelopes = 1

Total ways of NOT putting them correctly = 24 -1 = 23

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E?
4*4=16 ways.
Correct way = 1.
16-1=15

5. Good post? |
Originally Posted by rookie2005
None match the answer

Should be 23

Total ways of putting 4 letters in 4 envelopes (assuming an envelope can only hold one letter) = 4 * 3 * 2 *1 = 24

Total ways of putting all letters in their correct envelopes = 1

Total ways of NOT putting them correctly = 24 -1 = 23
I agree with this soln.

good one rookie.

6. Good post? |
Originally Posted by Ankost
E?
4*4=16 ways.
Correct way = 1.
16-1=15
Agree with Ankost.

Should be 15.

Total ways of putting 4 letters in 4 envelopes cant be 4!.

The first letter can be put in any one of the envelope in 4P1 ways.
Similary the 2nd letter can be put in any one of the envelope in 4P1 ways (and not 3P1)
3rd letter in 4P1 ways and the 4rth letter in 4P1 ways.

Total number of ways = 4+4+4+4 = 16.

7. Good post? |
Originally Posted by karmaholic
Agree with Ankost.

Should be 15.

Total ways of putting 4 letters in 4 envelopes cant be 4!.

The first letter can be put in any one of the envelope in 4P1 ways.
Similary the 2nd letter can be put in any one of the envelope in 4P1 ways (and not 3P1)
3rd letter in 4P1 ways and the 4rth letter in 4P1 ways.

Total number of ways = 4+4+4+4 = 16.
But once you put the first letter isn't there a reduced sample space???

8. Good post? |
The total number of ways of putting 4 letters into 4 envelopes can be

1) Case where an envelope can take any number of letters, ie some may be empty and others might have more than 1 = 4*4*4*4 = 4^4

2) Case where every envelope must have atleast one letter
= 4*3*2*1 = 24

Why is everyone adding 4 + 4 +4 +4 , Please explain

9. Good post? |
Ways to have no letter in its correct envelope = Total ways of enveloping - Number of ways to envelope atleast 1 letter correctly.

Total ways = 4! = 24

Correct - Incorrect:
1 - 3
Number of ways to envelope 3 letters incorrectly (all incorrect) = 2
=> total number of 1-3 combos = 4*2 = 8

2 - 2
Number of ways to envelope 2 letters incorrectly (all incorrect) = 1
=> total number of 2-2 combos = 4C2*1 = 6

4 - 0
Number of 4-0 combos = 1

Note: It's important to realize that a 3-1 (correct-incorrect) combo is impossible.

So, Ways to have no letter in its correct envelope = 24 - (8+6+1) = 9

10. Good post? |
very cool Arjmen!

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