Prove that the following identity about binomial coefficients` ((n),(0))+((n+1),(1))+((n+2),(2))+.....((n+r),(r))=((n+r+1),(r))`

.^(n-1)C_(r)+^(n-1)C_(r-1)=

A car is parked among N cars standing in a row, but not at either end. On his return, the owner finds that exactly "r" of the N places are still occupied. The probability that the places neighboring his car are empty is ((r-1)!)/((N-1)!) b. ((r-1)!(N-r)!)/((N-1)!) c. ((N-r)(N-r-1))/((N-1)(N+2)) d. (^(N-r)C_2)/(^(N-1)C_2)

Show that the sum of the coefficient of first (r+1) terms in the expansions of (1-x)^(-n) is ((n+1)(n+2)...(n+r))/(r!)

Show that , (.^(n)C_(r)+^(n)C_(r-1))/(.^(n)C_(r-1)+^(n)C_(r-2))=(.^(n+1)p_(r))/(r.^(n+1)p_(r-1))

Evaluate the following limit: lim_(nto oo)(sum_(r=1)^(n) sqrt(r)sum_(r=1)^(n)1/(sqrt(r)))/(sum_(r=1)^(n)r)

Find the coefficient of x^(r) in the following expression : (x+n)^(n)+(x+2)^(n-1)(x+1)+(x+2)^(n-2)(x+1)^(2)+.....+(x+1)^(n)

Find the sum sum_(r=1)^(n) r^(2) (""^(n)C_(r))/(""^(n)C_(r-1)) .

Probability if n heads in 2n tosses of a fair coin can be given by a. prod_(r=1)^n((2r-1)/(2r)) b. prod_(r=1)^n((n+r)/(2r)) c. sum_(r=0)^n(( ""^n C_r)/(2^n))^2 d. (sum_(r=0)^n(""^n C_r)^2)/(sum_(r=0)^(2n)(""^(2n)C_r)^2)

Prove that by using the principle of mathematical induction for all n in N : a+ ar+ ar^(2)+ ..+ ar^(n-1)= (a(r^(n)-1))/(r-1)

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