The total number of ways of selecting two numbers from the set `{1,2, 3, 4, ........3n}` so that their sum is divisible by 3 is equal to a. `(2n^2-n)/2` b. `(3n^2-n)/2` c. `2n^2-n` d. `3n^2-n`

For ninN , n^3+2n is divisible by

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

5^(2n+1)+3^(4n+2) (n in N) is divisible by

n is selected from the set {1, 2, 3,.............,100} and the number 2^n+3^n+5^n is formed. Total number of ways of selecting n so that the formed number is divisible by 4 is equal to (A) 50 (B) 49 (C) 48 (D) None of these.

Number of ways in which three numbers in A.P. can be selected from 1,2,3,..., n is a. ((n-1)/2)^2 if n is even b. n(n-2)/4 if n is even c. (n-1)^2/4 if n is odd d. none of these

If n (> 1)is a positive integer, then show that 2^(2n)- 3n - 1 is divisible by 9.

If n objects are arrange in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is a. "^(n-2)C_3 b. "^(n-3)C_2 c. "^(n-3)C_3 d. none of these

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 (2^(2n)-1) is divisible by 3, then show that [2^(2(n+1))-1] is also divisible by. 3

If (2^(2n)-1) is divisible by 3, then show that, [2^(2(n+1))-1] is also divisible by 3.