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

If ""^(n)P_r = ""^(n)P_(r-1) and ""^(n)C_(r) = ""^(n)C_(r-1) , find the values of n and r.

Prove that .^(n)C_(0) - ^(n)C_(1) + .^(n)C_(2)- ^(n)C_(3) + "…" + (-1)^(r) + .^(n)C_(r) + "…" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that .^(n)C_(1) + 2 .^(n)C_(2) + 3 .^(n)C_(3) + "…." + n . ^(n)C_(n) = n 2^(n-1) .

Prove ""^(n)P_(r)=""^(n-1)P_(r) +r. ""^(n-1)P_(r-1)

If ""^(n)C_(r) denotes the number of combinations of n things taken r at a time, then the expression ""^(n)C_(r+1)+""^(n)C_(r-1)+2xx""^(n)C_(r) , equals

Assertion (A) : ""_(n)P_(r) gt ""_(n)C_(r) Reason ""_(n)P_(r) = ""_(n)C_(r) xx r!

Prove that ""^(n-1)P_r+r^(n-1)p_r-1=^nP_r dot""

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

If ""^(nP_(r)=""^(n)P_(r+1) and ""^(n)C_(r)=""^(n)C_(r-1) find n and r.