Prove that `""^(n-2)C_r+2* ""^(n-2)C_(r-1)+ ""^(n-2)C_(r-2)=""^nC_r`

Prove that: ""^(n-1)P_r=(n-r)* ""^(n-1)P_(r-1)

Prove that ""^nC_r+""^nC_(r+1)+""^(n+1)C_(r+2)=""^(n+2)C_(r+2)

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

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

Prove that .^(n+1)C_(r+1)+^nC_r+^nC_(r-1)=^(n+2)C_(r+1)

Prove that: ""^(n+1)P_(r+1)=(n+1)* ""^nP_r .

Show that .^nC_r+.^(n-1)C_(r-1)+.^(n-1)C_(r-2)=.^(n+1)C_r

Show that ^nC_r+2. ^nC_(r-1)+ ^nC_(r-2)= ^(n+2)C_r

Without expanding show that : |(""^xC_r,""^xC_(r+1),""^xC_(r+2)),(""^yC_r,""^yC_(r+1),""^yC_(r+2)),(""^zC_r,""^zC_(r+1),""^zC_(r+2))|=|(""^xC_r,""^(x+1)C_(r+1),""^(x+2)C_(r+2)),(""^yC_r,""^(y+1)C_(r+1),""^(y+2)C_(r+2)),(""^zC_r,""^(z+1)C_(r+1),""^(z+2)C_(r+2))|

Prove one of the following: ^nC_r+^nC_(r-1)=^(n+1)C_r .