zura wrote:
How many arrangements of the letters of the word DEFEATED are there in which the three E are separated?
Take the case with constraints placed at the leftmost
E-E-E---
The four blank places can be filled in 5!/2=60 ways.
1. The last E can be moved 3 more positions to the right in addition to its present position , each giving 60 ways for a total of 60*4=240 ways
2. Now push the next rightmost E also to the right. We have 60*3=180 ways. Pushing the two rightmost E's can continue till 60*2 and 60*1 ways i.e, a total of 360 ways
3. Pushing all the E's by 1 position, we have 60*3 and 60*2 and 60*1 i.,e 360 ways ways . Pushing all the three E's can continue till 60*2 and 60 *1 and 60*1 ways i.e, a total of 600 ways
The grand total is 240+360+600=1200
As a short cut
step 1 is 60*4
Step 1 and 2 is 60*4 + (60*3 + 60*2 +60*1)
Step 1 ,2 and 3 is 60*4 + 60*3 + 60*2 +60 *1 + (60*3+60*2 +60*1) + (60*2+60*1) + (60*1)=1200
Thus these problems where we can place the constraints starting leftmost possible can be easily solved just by solving step 1.
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Srinivasan Vaidyaraman
Magical LogiciansHolistic and Holy Approach