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Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is answer:53

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5 years ago Latika Leekha
165 Points
```							Hello student,Number of derrangements of 6 = 6! (1 – 1/1! + 1/2! - 1/3! + 1/4! – 1/5! + 1/6!) = 360 – 120 + 30 – 6 + 1 = 265Out of these derrangements, there are five ways in which card numbered 1 is going wrong.So, when it is going in envelope numbered 2 is 265/5 = 53 ways.
```
5 years ago
```							Our interest is in finding |T| whereT = {f ∈ S : f(i) is not equal to = i, for i = 2, 3, 4, 5, 6}We classify the functionsin T into types depending upon whether f(2) = 1 or f(2) is not = 1. So, letT1 = {f ∈ T : f(2) = 1}                                (1)and T2 = {f ∈ T : f(2) is not = 1}                 (2)A function in T1 interchanges 1 and 2 and maps the remaining symbols3, 4, 5, 6 to themselves bijectively, without any fixed points. So it islike a derangement of these four symbols. Hence|T1| = D4 = 9                                           (3)as calculated above. Now consider a function f in T2. This correspondsto a bijection from the set {2, 3, 4, 5, 6} to the set {1, 3, 4, 5, 6} in whichf(2) is not = 1 and f(i) is not = i for i = 3, 4, 5, 6. If we relabel the element 1in the codomain as 2 (because we thought that 2 not to go into 1), then f is nothing but a derangement of the fivesymbols 2, 3, 4, 5 and 6. Therefore|T2| = D5 = 60 − 20 + 5 − 1 = 44              (4)Adding (3) and (4) we get|T| = |T1| + |T2| = 9 + 44 = 53
```
3 years ago
```							Hello friend,Number of derrangements of 6 = 6! (1 – 1/1! + 1/2! - 1/3! + 1/4! – 1/5! + 1/6!)= 360 – 120 + 30 – 6 + 1= 265Out of these derrangements, there are five ways in which card numbered 1 is going wrong.So, when it is going in envelope numbered 2 is 265/5 = 53 ways.Hope this may help you....
```
2 months ago
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