Let `f(n ,k)` denote the number of ways in which `k` identical balls can be colored with `n` colors so that there is at least one ball of each color. Then `f(2n ,n)` must be equal to a. `^2n C_n` b. `^2n-1C_(n+1)` c. `^2n-1C_n` d. none of these

The total number of ways in which n^2 number of identical balls can be put in n numbered boxes (1,2,3,.......... n) such that ith box contains at least i number of balls is a. .^(n^2)C_(n-1) b. .^(n^2-1)C_(n-1) c. .^((n^2+n-2)/2)C_(n-1) d. none of these

If N denotes the number of ways of selecting r objects of out of n distinct objects (rgeqn) with unlimited repetition but with each object included at least once in selection, then N is equal is a. .^(r-1)C_(r-n) b. .^(r-1)C_n c. .^(r-1)C_(n-1) d. none of these

There are (n+1) white and (n+1) black balls,each set numbered 1to n+1. The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colors is a.(2n+2)! b.(2n+2)!xx2 c.(n+1)!xx2 d.2{(n+1)!}^(2)

The total number of ways in which 2n persons can be divided into n couples is a.(2n!)/(n!n!) b.(2n!)/((2!)^(3)) C.(2n!)/(n!(2!)^(n)) d.none of these

There are two bags each of which contains n balls.A man has to select an equal number of balls from both the bags.Prove that the number of ways in which a man can choose at least one ball from each bag is ^(2n)C_(n)-1.

There are n idstinct white and n distinct black ball.If the number of ways of arranging them in a row so that neighbouring balls are of different colors is 1152, then value of n is

Two players P_1a n dP_2 play a series of 2n games. Each game can result in either a win or a loss for P_1dot the total number of ways in which P_1 can win the series of these games is equal to a. 1/2(2^(2n)-^ "^(2n)C_n) b. 1/2(2^(2n)-2xx^"^(2n)C_n) c. 1/2(2^n-^"^(2n)C_n) d. 1/2(2^n-2xx^"^(2n)C_n)

If A=[(1,1),(1,0)] and n epsilon N then A^n is equal to (A) 2^(n-1)A (B) 2^nA (C) nA (D) none of these