+1 vote
232 views
in how many ways 6 letters can be placed in 6 envelopes such that at least 4 letters go into their corresponding envelopes ?

edited | 232 views
0
22?
+1
16?
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30 ?
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Yeah! it should be 30
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Explanation?
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First we will select 4 letters from 6 and assign them to their respective envelopes. i.e. $^4C_2$

Then for remaining 2 letters there are 2! ways

Therefore 30
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Atleast 4 in right place.

So, selecting 4 out of 6 in 6C4 ways.

remaining two can come in two ways either arranged or not.

6C4 * 2 = 15*2 = 30
+3
You forgot to apply inclusion exclusion principle.

Suppose u put letter 1,2,3,4 into their respective envelopes now u r remaining with letter 5 and 6 which can be placed in Envelop 6 and 5 only. Means only one choice l. So it will be C(6,4). And the last one when all are in there respective envelopes.

Hence total should be 15+1=16.
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Bro, atleast 4 right? it is not atmost 4
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@Shubhanshu Yes, CORRECT! The answer should be $16$. Thanks.

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Yes muthu Kumar
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@Shubhanshu

Thanks bro. understood

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@muthu kumar Answer is not 30 bcoz, when you do $^4C_2$ and select say 1,2,3,4, then according to you there are 2 combinations,.. 1,2,3,4,5,6 and 1,2,3,4,6,5

Now if you select say 3,4,5,6 then also according to you there are 2 combinations..  1,2,3,4,5,6 and 2,1,3,4,5,6

Look here 1,2,3,4,5,6 is repeating..

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Yeah bro. Understood.

Exactly 4 in correct place = 6C4 * D2 = 15

Exactly 5 in correct place = 6C5 * D1 = 6

Exactly 6 in ocrrect place = 1

Total no of ways = 22
by (149 points)
+1

When you consider the case of exactly 5 in correct places, the 6th one is automatically in its correct place.

Therefore, 15+1=16.
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If we have 5 letters in correct places, then 6th one has only 1 place to go, which is its correct place. So answer should be 16

+1 vote