Maths-
General
Easy

Question

There were two women participating in a chess tournament Every participant played two games with the other participants the number of games that the men played between themselves proved to exceed by 66 the number of games that the men played with the women The number of participants is

  1. 6
  2. 11
  3. 13
  4. 10

The correct answer is: 13


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    If alpha = mC2, then alphaC2 is equal to -

    We are given that, 
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    We will expand the expression by using the identity 
    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
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    rightwards double arrow alpha space equals space fraction numerator m left parenthesis m minus 1 right parenthesis left parenthesis m minus 2 right parenthesis factorial over denominator 2.1 left parenthesis m minus 2 right parenthesis factorial end fraction

rightwards double arrow space alpha space equals space fraction numerator m left parenthesis m minus 1 right parenthesis over denominator 2 end fraction

    We have to find the value of C presuperscript alpha subscript 2 which is also equal to space fraction numerator alpha factorial over denominator 2 factorial left parenthesis alpha minus 2 right parenthesis factorial end fraction

    Also, we can rewrite it as 
    rightwards double arrow space fraction numerator alpha left parenthesis alpha minus 1 right parenthesis left parenthesis alpha minus 2 right parenthesis factorial over denominator 2.1 left parenthesis alpha minus 2 right parenthesis factorial end fraction

rightwards double arrow space space fraction numerator alpha left parenthesis alpha minus 1 right parenthesis over denominator 2 end fraction
    On substituting the values of alpha, we will get,
    fraction numerator begin display style fraction numerator m left parenthesis m minus 1 right parenthesis over denominator 2 end fraction end style open parentheses begin display style fraction numerator m left parenthesis m minus 1 right parenthesis over denominator 2 end fraction end style close parentheses over denominator 2 end fraction space equals space fraction numerator m left parenthesis m minus 1 right parenthesis left parenthesis m. left parenthesis m minus 1 right parenthesis minus 2 right parenthesis over denominator 8 end fraction
    Solve the brackets

    rightwards double arrow fraction numerator left parenthesis m squared space minus space m right parenthesis left parenthesis m squared space minus space m space minus 2 right parenthesis over denominator 8 end fraction
rightwards double arrow fraction numerator left parenthesis m squared space minus space m right parenthesis left parenthesis m squared space minus space 2 m space plus space m minus 2 right parenthesis over denominator 8 end fraction
rightwards double arrow fraction numerator left parenthesis m left parenthesis m minus 1 right parenthesis left parenthesis m left parenthesis space m minus 2 right parenthesis space plus 1 left parenthesis m minus 2 right parenthesis right parenthesis over denominator 8 end fraction
rightwards double arrow fraction numerator left parenthesis m minus 2 right parenthesis left parenthesis m minus 1 right parenthesis m left parenthesis m space plus 1 right parenthesis over denominator 8 end fraction
    Multiply and divide by left parenthesis m minus 3 right parenthesis factorial
    rightwards double arrow fraction numerator left parenthesis m minus 2 right parenthesis left parenthesis m minus 1 right parenthesis m left parenthesis m space plus 1 right parenthesis left parenthesis m minus 3 right parenthesis factorial over denominator 8 left parenthesis m minus 3 right parenthesis factorial end fraction space equals space fraction numerator left parenthesis m plus 1 right parenthesis factorial over denominator 8 left parenthesis m minus 3 right parenthesis factorial end fraction

    We can write m-3 as ((m+1) - 4)

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    If alpha = mC2, then alphaC2 is equal to -

    Maths-General
    We are given that, 
    alpha space equals space C presuperscript m subscript 2

    We will expand the expression by using the identity 
    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
    Hence, alpha space equals space fraction numerator m factorial over denominator 2 factorial left parenthesis m minus 2 right parenthesis factorial end fraction
    It is also known that
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    Then,
    rightwards double arrow alpha space equals space fraction numerator m left parenthesis m minus 1 right parenthesis left parenthesis m minus 2 right parenthesis factorial over denominator 2.1 left parenthesis m minus 2 right parenthesis factorial end fraction

rightwards double arrow space alpha space equals space fraction numerator m left parenthesis m minus 1 right parenthesis over denominator 2 end fraction

    We have to find the value of C presuperscript alpha subscript 2 which is also equal to space fraction numerator alpha factorial over denominator 2 factorial left parenthesis alpha minus 2 right parenthesis factorial end fraction

    Also, we can rewrite it as 
    rightwards double arrow space fraction numerator alpha left parenthesis alpha minus 1 right parenthesis left parenthesis alpha minus 2 right parenthesis factorial over denominator 2.1 left parenthesis alpha minus 2 right parenthesis factorial end fraction

rightwards double arrow space space fraction numerator alpha left parenthesis alpha minus 1 right parenthesis over denominator 2 end fraction
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    fraction numerator begin display style fraction numerator m left parenthesis m minus 1 right parenthesis over denominator 2 end fraction end style open parentheses begin display style fraction numerator m left parenthesis m minus 1 right parenthesis over denominator 2 end fraction end style close parentheses over denominator 2 end fraction space equals space fraction numerator m left parenthesis m minus 1 right parenthesis left parenthesis m. left parenthesis m minus 1 right parenthesis minus 2 right parenthesis over denominator 8 end fraction
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rightwards double arrow fraction numerator left parenthesis m squared space minus space m right parenthesis left parenthesis m squared space minus space 2 m space plus space m minus 2 right parenthesis over denominator 8 end fraction
rightwards double arrow fraction numerator left parenthesis m left parenthesis m minus 1 right parenthesis left parenthesis m left parenthesis space m minus 2 right parenthesis space plus 1 left parenthesis m minus 2 right parenthesis right parenthesis over denominator 8 end fraction
rightwards double arrow fraction numerator left parenthesis m minus 2 right parenthesis left parenthesis m minus 1 right parenthesis m left parenthesis m space plus 1 right parenthesis over denominator 8 end fraction
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    space fraction numerator left parenthesis m plus 1 right parenthesis factorial over denominator 8 left parenthesis left parenthesis m plus 1 right parenthesis space minus space 4 right parenthesis factorial end fraction

    Multiply and divide by 3 to make expression 4! in the denominator
    space fraction numerator 3 left parenthesis m plus 1 right parenthesis factorial over denominator 4 factorial left parenthesis left parenthesis m plus 1 right parenthesis space minus space 4 right parenthesis factorial end fraction space equals space C presuperscript 3 m plus 1 end presuperscript subscript 4





     
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