The number of ways to put 6 letters in 6 addressed envelopes so that all are in wrong envelopes?

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265

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264

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Solution

The correct option is C
265

Number of possible de arrangements that are possible out of 6 letters in 6 envelopes =

The number of ways without restriction - The number of ways in which all are in correct envelopes - The number of ways in which 1 letter is in correct envelope - The number of way in which 2 letters is in correct envelopes - The number of ways in which 3 letters is in correct envelopes - The number of way in which 4 letters is in correct envelopes - The number of ways in which 5 letter is in correct envelopes

= $6! - 1 -^{6} C_{1} \times$ Derangements of 5 letters - $^{6} C_{2} \times$ Derangements of 4 letters - $^{6} C_{3} \times$ Derangements of 3 letter - $^{6} C_{4} \times$ Derangements of 2 letters- $^{6} C_{5} \times$ Derangements of 1 letter
= $6! - 1 -^{6} C_{1} \times 44 -^{6} C_{2} \times 9 -^{6} C_{3} \times 2 -^{6} C_{4} \times 1 -^{6} C_{5} \times 0$
= 720 - 264 - 135 - 40 -15 = 265

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