Verify that `C(n,r) = n/r C(n-1,r-1)` and hence, prove that `C(n,r) = (lfloorn)/(lfloorrlfloorn-r)`

Verify that ""^nC_r= frac (n)(r) "^(n-1)C_(r-1) and hence prove that "^nC_r= (n!)/(r!(n-r)!) .

Evaluate (n!)/(r!(n - r)! , when n = 7 , r = 5

Prove that "^nC_r ^rC_5= ^nC_5 ^(n-5)C_(r-5) .

If t_r is the rth term is the expansion of (1 + a)^n , in ascending powers of a, prove that : r(r+1) t_(r+2)= (n-r+1)(n-r) a^2 t_r .

Evaluate (n!)/(r!(n - r)! , when n = 10 , r = 2

Evaluate (n!)/((n-r)!) , when n= 6, r= 2

Evaluate (n!)/((n-r)!) , when n= 9,r= 5.

Let P(n) be the statement ‘‘4^n > n’’ . If P(r) is true, prove that P(r +1) is also true.

P (n) is the statement ''2^(n)>3n'' and if P(r) is true, then prove that P(r + 1) is also true, n in N.