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Permutations And Combinations

Question
CBSEENMA11014210

If straight C presuperscript straight n minus 1 end presuperscript subscript straight r colon straight C presuperscript straight n subscript straight r colon straight C presuperscript straight n plus 1 end presuperscript subscript straight r space equals space 6 colon 9 colon 13 comma Find n and r.

Solution
straight C presuperscript straight n minus 1 end presuperscript subscript straight r colon straight C presuperscript straight n subscript straight r space equals space 6 colon 9
space space straight C presuperscript straight n subscript straight r plus straight C presuperscript straight n minus 1 end presuperscript subscript straight r space equals space 9 over 6
rightwards double arrow         space space space space fraction numerator straight n factorial over denominator left parenthesis straight n minus straight r right parenthesis factorial space straight r factorial end fraction cross times fraction numerator straight r factorial space left parenthesis straight n minus 1 minus straight r right parenthesis factorial over denominator left parenthesis straight n minus 1 right parenthesis factorial end fraction equals 3 over 2

rightwards double arrow                       fraction numerator straight n factorial over denominator left parenthesis straight n minus straight r right parenthesis factorial end fraction cross times fraction numerator left parenthesis straight n minus straight r minus 1 right parenthesis factorial over denominator left parenthesis straight n minus 1 right parenthesis factorial end fraction equals 3 over 2                
rightwards double arrow                      fraction numerator straight n over denominator straight n minus straight r end fraction equals 3 over 2
rightwards double arrow                            2n = 3n - 3r
rightwards double arrow                         n - 3r =   0                                                  ...(i)
Also,              straight C presuperscript straight n plus 1 end presuperscript subscript straight r plus straight C presuperscript straight n subscript straight r equals 13 over 9
rightwards double arrow          fraction numerator left parenthesis straight n plus 1 right parenthesis factorial over denominator straight r factorial left parenthesis straight n plus 1 minus straight r right parenthesis factorial end fraction cross times fraction numerator left parenthesis straight n minus straight r right parenthesis factorial space straight r factorial over denominator straight n factorial end fraction space equals space 13 over 9
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#6 {main}</pre>
 
rightwards double arrow                                      9n + 9 = 13n - 13r + 13
rightwards double arrow                                    4n - 13r = -4                                    ...(ii)
Multiplying (i) by 4, we get
                                       4n - 12r = 0                                       ...(iii)
Subtracting (iii) from (ii), we get          -r = -4  rightwards double arrow r = 4
From (i),                               n - 12 = 0
rightwards double arrow                                             n = 12
Hence, n = 12, r = 4.

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