`C_0-C_1+C_2-C_3+......+(-1)^rC_r=((-1)^r(n-1)!)/(r!.(n-r-1)!)`

C_(0)-C_(1)+C_(2)-C_(3)+......+(-1)^(r)C_(r)=((-1)^(r)(n-1)!)/(r!*(n-r-1)!)

Prove that (r+1)^(n)C_(r)-r^(n)C_(r)+(r-1)^(n)C_(2)-^(n)C_(3)+...+(-1)^(r)n_(C_(r))=(-1)^(r_(n-2))C_(r)

If 1<=r<=n, then n^(n-1)C_(r)=(n-r+1)^(n)C_(r-1)

If C_(r) = ""^(n)C_(r) and (C_(0) + C_(1)) (C_(1) + C_(2)) … (C_(n-1) + C_(n)) = k ((n +1)^(n))/(n!) , then the value of k, is

Prove that C_(0)2^(2)C_(1)+3C_(2)4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)C_(n)=0 where C_(r)=nC_(r)

The number of ways in which n different thingscan be distributed into r different groups is r^(n)-r_(1)(r-1)^(n)+^(r)C_(2)(r-2)^(n)-....+(-1)^(r-1)*rC_(r-1) or sum_(p=0)^(r)(-1)^(p)*rC_(p)*(r-p)^(n)

If (1 + x)^(n) = sum_(r=0)^(n) C_(r) x^(r),(1 + (C_(1))/(C_(0))) (1 + (C_(2))/(C_(1)))...(1 + (C_(n))/(C_(n-1))) is equal to