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Asked by krrishgarg967 | 07 Jan, 2023, 23:31: PM

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begin mathsize 14px style P presuperscript n subscript r space equals space scriptbase p subscript r plus 1 end subscript end scriptbase presuperscript n end style

begin mathsize 14px style fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction space equals space fraction numerator n factorial over denominator left parenthesis n space minus space r space minus 1 space right parenthesis factorial end fraction end style

begin mathsize 14px style fraction numerator left parenthesis n minus r minus 1 right parenthesis factorial over denominator left parenthesis space n space minus r space space right parenthesis factorial end fraction space equals space fraction numerator 1 over denominator left parenthesis n minus r right parenthesis end fraction space equals space 1 end style

Hence ( n-r) = 1

begin mathsize 14px style C presuperscript n subscript r space equals space C presuperscript n subscript r minus 1 end subscript end style

begin mathsize 14px style fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial space r factorial end fraction space equals space fraction numerator n factorial over denominator left parenthesis n minus r plus 1 right parenthesis factorial space left parenthesis r minus 1 right parenthesis factorial end fraction end style

if we use the substitution (n-r) = 1, then we get

begin mathsize 14px style 2 space cross times space fraction numerator left parenthesis space r space minus 1 space right parenthesis factorial over denominator r space factorial end fraction space equals space 2 over r space equals space 1 end style

i.e., r = 2

if ( n-r) = 1 , then n = 3

Answered by Thiyagarajan K | 08 Jan, 2023, 14:07: PM

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