(A) The number of ways of not selecting (n−1) things from n different things =nCn−r
(B) (n−r+1)⋅nCr−1 =(n−r+1)×
n!
(r−1)!(n−r+1)!
=
(n−r+1)×n!
(r−1)!(n−r+1)(n−r)!
=
n!
(r−1)!(n−r)!
=r
n!
r!(n−r)!
=r⋅nCr
(C) The number of ways of selecting atleast (n−r) things from n different things =nCn−r+nCn−r+1+nCn−r+2+...+nCn =nCn+nCn−1+nCn−2+...+nCn−r+1+nCn−r =1+nC1+nC2+...+nCr