The greatest value of 2nCr(0  ≤  r  ≤  2n)  is

We have 2nCr  −  12nCr=(2n)!(r−1)!(2n−r+1)!.r!(2n−r)!(2n)!                                 =r2n−r+1 For 1  ≤  r  ≤  n.   r2n−r+1<1.  Therefore, 2nC02nC1<1,  2nC12nC2<1,...  ,2nCn−12nCn<1, ⇒     2nC0<2nC1<2nC2<...<2nCn−1<2nCn Since 2nC2n−r=2nCr,  it follows that          2nC2n<2nC2n−1<2nC2n−2<...<2nCn + 1<2nCn Thus, the maximum value of 2nCr  is 2nCn.