Prove that: `n(n-1)(n-2)....(n-r+1)=(n !)/((n-r)!)`

Prove that: (i) (n!)/(r!) = n(n-1) (n-2)......(r+1) (ii) (n-r+1). (n!)/((n-r+1)!) = (n!)/((n-r)!)

Prove that (n!)/(r!(n-r)!)+(n!)/((r-1)!(n-r+1)!) =((n+1)!)/(r!(n-r+1)!)

If ((n),(r )) "denotes " ""^nC_r then (a) Evalutae : 2^(15)((30),(0))((30),(1))-2^(14)((30),(1))((29),(14))+2^(13)((30),(2))((28),(13)).......-((30),(15))((15),(0)) ( b) Prove that : Sigma_(r=1)^(n) ((n-1),(n-r))((n),(r))=((2n-1),(n-1)) ( c) Prove that : ((n),(r))((r),(k))=((n),(k))((n-r),(r-k))

Prove that P(n,r) = (n- r+1) P(n,r-1)

Prove that ^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r

Prove that .^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r

Prove that .^(n)C_(r )+.^(n-1)C_(r )+..+.^(r )C_(r )=.^(n+1)C_(r+1)

Prove that: (i) r.^(n)C_(r) =(n-r+1).^(n)C_(r-1) (ii) n.^(n-1)C_(r-1) = (n-r+1) .^(n)C_(r-1) (iii) .^(n)C_(r)+ 2.^(n)C_(r-1) +^(n)C_(r-2) =^(n+2)C_(r) (iv) .^(4n)C_(2n): .^(2n)C_(n) = (1.3.5...(4n-1))/({1.3.5..(2n-1)}^(2))

Prove that : (""^(n)C_(r+1))/(""^(n)C_(r))=(n-r)/(r+1)

Prove that ((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(r))