Maths-
General
Easy

Question

The greatest possible number of points of intersections of 8 straight line and 4 circles is :

  1. 32    
  2. 64    
  3. 76    
  4. 104    

Hint:

The number point of intersection between two lines can be counted by finding the number of ways in which two lines can be selected out of the lot as two lines can intersect at most one point.
The number point of intersection between two circles can be counted by finding the number of ways in which two circles can be selected out of the lot multiplied by 2 as two circles can intersect at most two points.

The correct answer is: 104


     Detailed Solution
    The number point of intersection between two circles can be counted by finding the number of ways in which one circle and one line can be selected out of the lot multiplied by 2 as one circle and one line can intersect at most two points.

    For selecting r objects from n objects can be done by using the formula as follows
    C presuperscript n subscript r space equals space fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction
     As mentioned in the question, we have to find the total number of intersection points.
    For calculating the points of intersection between two lines, we can use the formula which is mentioned in the hint as follows = C presuperscript 8 subscript 2 space cross times 1 space equals space 28
     For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows = C presuperscript 4 subscript 2 space cross times 2 space equals space 12

     For calculating the points of intersection between one line and one circle, we can use the formula which is mentioned in the hint as follows = C presuperscript 4 subscript 1 space cross times C presuperscript 8 subscript 1 cross times 2 space equals space 64

     Hence, the total number of points of intersection is = 28 + 64 + 12 = 104

    The students can make an error if they don’t know about the formula for calculating the number of points as mentioned in the hint which is as follows
    The number point of intersection between two lines can be counted by finding the number of ways in which two lines can be selected out of the lot as two lines can intersect at most one point.
    The number point of intersection between two circles can be counted by finding the number of ways in which two circles can be selected out of the lot multiplied by 2 as two circles can intersect at most two points.
    The number point of intersection between two circles can be counted by finding the number of ways in which one circle and one line can be selected out of the lot multiplied by 2 as one circle and one line can intersect at most two points.

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