 आदव पच्छो और संटम खार्मी यूप चन्डल जहापे हम आपके लेग यह यह देन आपके लेग यह नहीं ऋए लेएई आपके लेए आपके लेएई आहूं तो हाती बेसिक्रे कुछ आसा दिकता है तो दोस तो हाती की जो चाल होती है वो होती है कि जिस रो या और जिस कुलम पूछिषन पे रही करता है चस बोट पे उनके सारे शकौयस को तो जो कनतोल करता है कुछ इस तरा याप देख पारे हो गय अपने फिगर में कि एक रुक को मैंने तरद रो आन सेवंत कुलम पूछिषन पे रहा है तो जो भी स्कौयस आप को या दिक रहे हैं यो की येल्लो कलर से आप को दर्षाय जा रहे हैं दोस हाप बिन फिल्ट विद दी येल्लो कलर देविल आल भी कन्तोल बाए दे रुक तो स्वने च्यस Eventually it has been played लोगो को पताहगी है की हा थी किस किष कौयस को कं reincarnटा है तो जिस रो क्य कोलम नेजा ईप पसिश अनtheme उस किस साथे कौईस को कॉन्oncesौल कर्तोल ब रहे है तो थो यह आप search here is asked की हमें व vicinity शबसे फय में what is the smallest number of rooks which can be arranged so that every square on the chess board is controlled by atleast one of them so those are the least number of rooks that would be required to control would be 8 which is the number of rows or the number of columns in the given chess board because if there is less than 8 then what will happen? one particular row will be like this or one particular column will be like this whose squares will not be controlled by any form so here the minimum number of rooks would be 8 so the first part of the question was very simple there is no work to be done in less than 8 there will be no square in any of the rows or columns which will not be controlled by any form if you take less than 8 then the number of rooks will be less now the second question is a very important question in how many ways can this be done i.e. how can you arrange your rooks in a chess board of 8 by 8 dimensions so that every square of the chess board is controlled by atleast one of them so friends we understand that we can arrange these rooks column wise or row wise and we will get the same result from both of them i.e. as an example, i have arranged the rooks in a row wise so as you can see in this animation that the first rook i have placed on the first row and i am also showing you which squares are being controlled by that particular rook so i have placed each rook in each row and here you will notice that all the squares of that rook and all the columns of that rook will be controlled by that rook so friends every rook as you can see our rooks are rook 1, rook 2, rook 3, dot dot dot till rook 8 every rook will have 8 options yes in any of the rows in any of the rows any 8 columns can be placed as you can see in this diagram in this animation your first rook is placed on the 7th column position similarly the second rook is placed on the 2nd row and it is placed on the 6th column position the 3rd rook is placed on the 5th column position and the 4th rook is placed on the 3rd column position so you will be able to see that the way you have placed it is not only controlling the squares of that rook but also controlling the respective columns so friends here every rook will have 8 options so in a particular row they can occupy any of the 8 column positions isn't it so if you are arranging the rooks row wise the total number of ways in which you can do is 8 to the power of 8 because here the fundamental principle of multiplication will be applied yes because you have to keep all the rooks and every rook has 8 options so your fundamental principle of multiplication is that the total number of ways to do it is 8 x 8 x 8 x 8 times which is 8 to the power of 8 I can say that column wise if you will keep them then the result will be the same so column wise will be 8 to the power of 8 so here the kids will feel that the total number of ways to do it will be just the sum of these 2 because these are 2 cases yes you will feel that this is case 1 and case 2 so all the number of ways to do case number 1 and all the number of ways to do case number 2 if you add them then our answer should be removed but wait friends here there will be a small mistake there will be a mistake because in your row wise and column wise arrangement of rooks some cases will be common yes yes what will be the common cases what will be the common cases so the common situation friends will be this situation where as you have kept rook number 1 here so rook number 2 will not come in this whole column of course it will not come in this row this means that rook number 1 had 8 options for you but rook number 2 which you are going to place it will have the same option because it will not be able to come in this position correct so assume you have kept rook number 2 here so rook number 3 will not be able to come in this whole column this means that rook number 3 will have 2 positions less means it will have only 6 positions here here here here and here only it can come similarly rook number 3 which you have placed here so whatever squares will come below this our rook number 4 will not come below that rook number 1 and rook number 2 already we have removed their columns so our rook number 4 will have only 5 options yes here here here and here only it can come similarly we can say that our last rook will be rook number 8 it will have only one option that means the number of common cases that you will end up getting will be 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 which is 8 factorial and these are the cases which will occur in your row wise and column wise that is why the final answer total number of ways would be 8 to the power 8 plus 8 to the power 8 minus the common cases so we will have to remove one common case because these 8 factorial will appear in our row wise and column wise in both the arrangements so our final answer of this question will be 2 x 8 to the power 8 minus 8 factorial so I hope you have learnt a lot from this video and this could be a probable question in the JEE Advanced examination thank you so much for watching stay safe, stay healthy