A circle is inscribed in a square with an area of 121 square

Well the official explaination is correct. we should take all cases unless specified. Here Bill did (7-1)! considering no difference in clockwise and anticlockwise. But we should take all options. I hope it is clear!

Area of square = 121 sq units. Hence side = sqrt(121) = 11 units.

As circle is inscribed in square, so the diameter = 11.

Circumference = 2π x 11/2 = 11π

Diameter of each marble = 1.5π
So maximum possible marbles possible on circumference = 1.5π x 7 = 10.5π < 11π

Number of ways to select 7 out of 9 possible marbles = 9C7 = 36.
To arrange these 7 marbles around circumference = (7 - 1 )! = 720.
Total arrangements = 36 x 720 = 25920.

I am not sure about the options because we can get option D as answer only when we take 36 x 7! = 181440 , instead of 36 x 6!= 25920.

Are you sure about the options ?

well here is the official explanation.



well my doubt is whether the official explanation provided by 800score is correct or incorrect. because i certainly believe that answer of this question should be 25920, as calculated by billgill in the previous post.

Do the marbles lie inside the circumference or outside the circumference ?
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In general when we count, we always have one fixed point of reference, with respect to which we count for example in Cartesian coordinate system we have origin as a reference and the naming of points is done with respect to origin. similarly on number line we have 0 as reference point. but in case of circle we don't have that reference point with us, that is why we always subtract 1 when we write total number of arrangements.

now lets look into the official explanation.



now what is the first spot. as i said earlier we need some point of reference to call any point as first. since that point of reference is missing, i believe we can't simply call any point as point one. hence i believe that this solution is incorrect.

GOT IT! answer should be 25920. once again kudos to bill and manpreet

This is indeed a 800 question. The way of calculating 25920 is absolutely correct but the actual answer is yet reached.

why the answer is 36 * 7! and not 36 * 6! ?

Here is the explanation:

In this case, the circle can accommodate only 7 marbles. However, when all the marbles are arranged, there will definitely be a gap so that all the marbles will not form a complete circle. (n-1)! can be valid only when the arrangement is complete and the circle is formed so that there is no starting or ending point.

In this case, the seven marbles can be arranged around the circle and the gap can come between any two marbles.

Let "|" be the gap and "o" be the marbles.

"|" can fit in ooooooo at seven places even when placed around in a circle.

Hence the 6! has to be multiplied with 7. This forms a general case of permutation


Hope it helps.

Kudos if you like the post

Bunuel can you pls post your explanation to this question.
Can anyone pls explain
Thanks

Square Area = 121 , Side s = 11
Circumference of a circle = 2pir = 2x11/2xpi = 11pi
Marble ball = 1.5pi
I do not understand how to use diameter of marble ball ??

+1 for answer incorrect. The answer would have been the official explanation only if the spots were numbered. This is convoluted wrong question that must be removed.

+1 for incorrect. I don't agree that we can count the space between marbles as a unit as above. This space could be infinitely variable under the above conditions.

I agree that the answer should be 25920.

Please either clarify or remove this.

We see that the side of the square is √121 = 11. Since the side of the square is equal to the diameter of the circle, we have the diameter = 11, and hence the circumference of the circle is 11π.

Since 11π/(1.5π) = 11/(3/2) = 22/3 = 7⅓, we can fit only 7 marbles around the circumference of the circle. Thus, there are 9C7 = 9C2 = (9 x 8)/2 = 36 ways to fit 7 of the 9 marbles around the circumference. However, for each set of 7 marbles that fit around the circumference, there are (7 - 1)! = 6! = 720 ways to arrange them. Therefore, there are 36 x 720 = 25,920 different combinations.

(Note: The official answer is D, which is only true if 36 is multiplied by 7! Instead of 6!. Notice 36 x 7! = 36 x 5040 = 181,440. But it’s a circular permutation problem, and so the answer should be 25,920.)
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Seems out there, but is it possible that the Square that the Circle is Inscribed in creates the Point of Reference such that it is not a true Circular Permutation (n-1)! ?

For instance, when you pick for the 1st Spot, in a Circular Permutation Each Slot is Identical because we have no frame of reference.

Here, if you Number the Spots 1 thru 7, you can say that Spot 1 is to the Left of the Upper Right Corner of the Square, etc. depending on how you draw the picture. This creates a Point of Reference for each Empty Seat around the Circle's Circumference at the beginning, and we can say they are Distinct Seats.

a reach, but....

Responding to a pm:

Note that the circle is inscribed in a square and we need to place 7 marbles around the circle. Though the placement is symmetric on the circle, with respect to the square, the placements are distinct.
Say point A is exactly middle of a side of the square. No other point will be exactly in the middle of the side of the square because we have 7 marbles.



So we consider each point unique and go with 9C7 * 7!

By the way, not a very high quality question. There is a fair bit of ambiguity and approximation.

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