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Consider two strings $A$="qpqrr" and $B$="pqprqrp". Let $x$ be the length of the longest common subsequence (not necessarily contiguous) between $A$ and $B$ and let $y$ be the number of such longest common subsequences between $A$ and $B$. Then $x +10y=$ ___.

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LCS of length 4 ==> x=4

LCS={qpqr,qprr,pqrr} ==>y=3

x+4y=34.

Answer is $34$.

In first string, if we want to get $4$ as maximum length then LCS should end with either "$rr$" or "$qr$".
Only $4$ combinations are possible for LCS with length $4$:

"$qpqr$"

"$qqrr$"

"$pqrr$"

"$qprr$"

Now, check for matching sequences in second string, except for "$qqrr$" all are possible.
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Do we always need to construct a table(Dynamic Programming Technique) ? or is there some other short method also...for calculating length of LCS?
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how do u get qqrr ?? is it in sequence ??
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we cant get qqrr..
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Why should we dont considr permutation of rwo differnt qpqr

That way answer would become 44
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plz clearify

Here in question it is explicitly mentioned (Not necessarily Contiguous) but if Only Longest Common Subsequence is mentioned what should be our approach?Will it be same???

I hope this helps. :)

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PLEASE Check in 5th row 4th colum their is a match of R  and you have not incremented it ... but still you are getting  through answer ... how ??
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Thank you @bharti ji. Typo is corrected.
Its a big problem if we solve it with DP or recursion.So we should use some intuition to get answer.

A=5 length and B= 7 length.

Maximum x can be 5 but 5 is not possible so try with 4 and we can easily find that 4 is satisfying for x.Now we need to find y.Here we know x=4 ,so let us write all the 4 length subsequences of A.As A is 5 length and we need 4 length so remove 1 character at a time and get the subsequence.

1. qpqr

2.qpqr

3.aprr

4.qqrr

5.pqrr

Now look for these in B and we will see except qqrr all are satisfying.So y=4 ,but 1 and 2 are same ,so y=3.

Now x+10y=34
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No, we can't qqrr...as common.../// whr u r facing [email protected]
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slight mistake ....
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Ok..
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Nice approach !! Although successful in certain cases only !
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Thanks rahul sharma 5 for such a nice approach

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There is $qprr$ in place of $aprr$ @rahul sharma 5

step 1- LCS(i,j) depends on LCS(i+1,j+1) when match else max{ LCS(i+1,j) and LCS(i,j+1)}

step 2- Starts at LCS(m,n) and fill by row,column or diagonal

step 3- When there is match then value of that cell 1+ LCS(i+1,j+1)  else step 1

Black Arrow - Fill columns and other arrow for backtracking

### x+10y = 4+10 * 3 = 34

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$pqrr,qpqr,qprr$ are three strings acc to figure
How to calculate the number of largest common subsequences..?
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This will help you to calculate longest sub sequences by drawing a table and from there it will be easier for you to figure out number of longest subsequence

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@Abhishek video is fine . But, still how to find the number of LCSs ??
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@VS.Check my answer.It might give a little help
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How

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