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Byju's Answer
Standard VIII
Mathematics
Construction of Parallel Lines
If m parallel...
Question

If m parallel lines in a plane are intersected by a family of n parallel lines , find the nunber of parallelograms formed

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Solution

Dear student

begin mathsize 20px style I f space m space p a r a l l e l space l i n e s space i n space a space p l a n e space a r e space i n t e r s e c t e d space b y space space a space f a m i l y space o f space n space p a r a l l e l space l i n e s t h e n space t h e space n u m b e r space o f space p a r a l l e log r a m s space f o r m e d space i s colon to the power of m C subscript 2 cross times to the power of n C subscript 2 equals fraction numerator m factorial over denominator 2 factorial open parentheses m minus 2 close parentheses factorial end fraction cross times fraction numerator n factorial over denominator 2 factorial open parentheses n minus 2 close parentheses factorial end fraction equals fraction numerator m open parentheses m minus 1 close parentheses open parentheses m minus 2 close parentheses factorial over denominator 2 open parentheses m minus 2 close parentheses factorial end fraction cross times fraction numerator n open parentheses n minus 1 close parentheses open parentheses n minus 2 close parentheses factorial over denominator 2 open parentheses n minus 2 close parentheses factorial end fraction space equals fraction numerator m open parentheses m minus 1 close parentheses up diagonal strike open parentheses m minus 2 close parentheses factorial end strike over denominator 2 up diagonal strike open parentheses m minus 2 close parentheses factorial end strike end fraction cross times fraction numerator n open parentheses n minus 1 close parentheses up diagonal strike open parentheses n minus 2 close parentheses factorial end strike over denominator 2 up diagonal strike open parentheses n minus 2 close parentheses factorial end strike end fraction space space space space space equals fraction numerator m n open parentheses m minus 1 close parentheses open parentheses n minus 1 close parentheses over denominator 4 end fraction F o r m u l a space u s e d colon 1 right parenthesis space n factorial equals n open parentheses n minus 1 close parentheses open parentheses n minus 2 close parentheses cross times... cross times 4 cross times 3 cross times 2 cross times 1 2 right parenthesis space to the power of p C subscript q equals fraction numerator p factorial over denominator q factorial open parentheses p minus q close parentheses factorial end fraction end style
Regards

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