PERMUTATIONS AND COMBINATIONS | PERMUTATION UNDER CERTAIN CONDITIONS, PERMUTATION OF OBJECT NOT ALL DISTINCT, COMBINATION | Prove that the no. of all permutations of n different objects taken r at a time when a particular object is to be always included in each arrangement is `r.(n-1)P_(r-1)`, Prove that number of permutations of n distinct objects taken r at a time ; when a particular object is never taken in each arrrangement is `(n-1)P_r`, Prove that the no. of permutations of n different objects taken r at a time in which two specified objects always occur together `2!(r-1) (n-2)P_(r-2)`, Prove that the no. of mutually distinguishable permutations of n things ; taken all at a time of which p are alike of one kind; q alike of second kind such that `p+q=n` is `(n!)/(p!.q!)`, Combinations intro & difference between combination and permutation, theorem:- The no. of all combinations of n distinct objects taken r at a time is given by `nC_r`= `(n!)/((n-r)!.r!)`, Property:- (i) `nC_r=nC_(n-r)` (ii) `(nC_r)/(r+1)=((n+1)C_(r+1))/(n+1)`, Property :- (iii) `nC_r+nC_(r-1)=(n+1)C_r`, Property (iv) If `^nC_x=^nC_y`; then either x=yx=y or x+y=nx+y=n (v) r.nC_r=n. (n-1)C_(r-1)r.nC_r=n. (n-1)C_(r-1), Property:Product of r consecutive number is divisible by r!