n different books `(n>=3)`are put at random in a shelf.Among these books there is a particular book A and a particular book B.The probability that there are exactly r books between A and B is `(A) (2)/(n(n-1)) ` `(B) (2(n-r-1))/(n(n-1)` `(C) (2(n-r-2))/(n(n-1))` `(D) (n-r)/(n(n-1)`

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

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

(ii) (n!)/((n-r)!r!)+(n!)/((n-r+1)!(r-1)!)=((n+1)!)/(r!(n-r+1)!)

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

A car is parked among N cars standing in a row,but not at either end.On his return,the owner finds that exactly r of the N places are still occupied.The probability that the places neighboring his car are empty is a.((r-1)!)/((N-1)!) b.((r-1)!(N-r)!)/((N-1)!) c.((N-r)(N-r-1))/((N-1)(N+2)) d.((N-r)C_(2))/(N-1)C_(2)

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

Show that the number of ways in which n books may be arranged on a shelf so that two particular books shall not be together is (n-2)(n-1)!

Ifn>2than sum_(r=0)^(n)(-1)^(r)(n-r)(n-r+1)C_(r)=(A)0(B)n(C)2^(n)(D)(n-1)2^(n)

If sum_(r=1)^(n)r^(3)((C(n,r))/(C(n,r-1)))=14^(2) then n=