Question

If nPr = nPr+1 and nCr= nCr-1, find the values of n and r.

Answer 1

ⁿPr=ⁿPr+1
or, n!/(n-r)!=n!/(n-r-1)!
or, n!/(n-r)(n-r-1)!=n!/(n-r-1)!
or, 1/(n-r)=1
or, n-r=1 ------------------(1)
Again,
ⁿCr=ⁿCr-1
or, n!/r!(n-r)!=n!/(r-1)!(n-r+1)!
or, 1/r(r-1)!(n-r)!=1/(r-1)!(n-r+1)(n-r)!
or, 1/r=1/(n-r+1)
or, r=n-r+1
or, n-2r=-1 --------------------(2)
Subtracting (2) from (1) we get,
n-n-r+2r=1+1
or, r=2
Putting in (1),
n-2=1
or, n=3
∴, n=3, r=2