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Assume that there are 6 color letters L1,L2,L3,L4,L5,L6  are to be placed in 6 same color envelope E1,E2,E3,E4,E5,E6(one letter for each envelop). The number of possibilities to place exactly one letter in the correct envelop______.

a)264

b)265

c)455

d)466
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apply dearrangments here.

first place 1 letter in correct position and find dearrangments for remaining 5 letters . this procedure must be repeated for all 6 individual correct places

letter1          
ENV1 ENV2 ENV3 ENV4 ENV5 ENV6

 

as letter 1 is at right position so find dearrangements for remaining 5 letters

i.e. 5!-$\binom{5}{1}$*4!+$\binom{5}{2}$*3!-$\binom{5}{3}$*2!+$\binom{5}{4}$*1!-1

which comes out to be 44.

for 6 individual correct positions we have

6*44=264 possibilities

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