document.write( "Question 1014498: Show that nPr = nPr+1 (show that n permutation r equals to n permutation r+1). \n" ); document.write( "
Algebra.Com's Answer #630779 by mathmate(429)\"\" \"About 
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\n" ); document.write( "Question:
\n" ); document.write( "Show that nPr = nPr+1 (show that n permutation r equals to n permutation r+1)
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\n" ); document.write( "Solution:
\n" ); document.write( "We will use the notation \"P%28n%2Cr%29=nPr=n%21%2F%28n-r%29%21\", so that
\n" ); document.write( "if and when
\n" ); document.write( "P(n,r)=P(n,r+1), then
\n" ); document.write( "P(n,r)-P(n,r+1)=0.............(1)
\n" ); document.write( "expanding above
\n" ); document.write( "\"n%21%2F%28n-r%29%21-n%21%2F%28n-%28r%2B1%29%29%21\"=0
\n" ); document.write( "\"n%21%2F%28n-r%29%21-n%21%2F%28n-r-1%29%21\"=0
\n" ); document.write( "Add by cross multiplication:
\n" ); document.write( "\"%28%28n%21%29%5E2%2A%28n-r%29%21%2A%28n-r-1%29%21%29%2F%28%28n-r%29%21%2A%28n-r-1%29%21%29\"=0.....(1a)
\n" ); document.write( "(1a) can be satisfied if and only if the numerator equals zero.
\n" ); document.write( "=>
\n" ); document.write( "n=0, trivial solution if r=0.
\n" ); document.write( "n=r, leads to (-1)! in denominator, rejected
\n" ); document.write( "n-r-1=0, means n=r+1
\n" ); document.write( "Thus
\n" ); document.write( "The above equation can be satisfied when n=r+1, or
\n" ); document.write( "P(n,n-1)=P(n,n) for all n>0.
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\n" ); document.write( "Note: if there is a typo in the original question, please post a new question.
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