If the number of$seven-digit$numbers, such that the sum of their digits is even, is m$\cdot n$$\cdot 10$; n m, n$\i_n {1, 2, 3,$$\ldots, 9}, then$m + n$is equal to _______$
For n$\ge 2, let S denote the set of all subsets of {1, 2$$\ldots..., n} with no two consecutive numbers. For example n {1, 3, 5}$$\i_n$$S_{6}$$, but {1, 2, 4}$$\notin$ S. Then n ( S ) is equal to ________ 6 5
Let $N$ be the number of four digit even numbers such that if 3 is one of the digits, then 5 is the succeeding digit. If $N = P_1^\alpha P_2^\beta P_3^\gamma$ ($P_1,P_2,P_3$ are primes, $\alpha,\beta,\gamma\in\mathbb{N}$), then $P_1+P_2+P_3$ is
Let $N$ be the number of four digit even numbers such that if 3 is one of the digits, then 5 is the succeeding digit. If $N = P_1^\alpha P_2^\beta P_3^\gamma$ ($P_1,P_2,P_3$ are primes, $\alpha,\beta,\gamma\in\mathbb{N}$), then $P_1+P_2+P_3$ is
Other than the letter S, the seven letters M, I, I, I, P, P and I can be arranged in \(\dfrac{7!}{2! \cdot 4!} = 7 \cdot 5 \cdot 3\). Now four S can be placed in eight spaces in \({}^8C_4\) ways. Therefore, the required number of ways is \(7 \cdot 5 \cdot 3 \cdot {}^8C_4 = 7 \cdot {}^6C_4 \cdot {}^8C_4\). The required number of ways is:
Consider three boxes, each containing 10 balls labelled 1, 2, ..., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by \(n_i\) the label of the ball drawn from the \(i^{\text{th}}\) box, \((i = 1, 2, 3)\). Then, the number of ways in which the balls can be chosen such that \(n_1
275. \(ABCD\) is a rectangle with vertices \(A(0,0)\), \(B(m,0)\), \(C(m,n)\) and \(D(0,n)\) with \(m, n \in N\). Points are chosen starting from \(A_0(0,0) \to A_1(x_1,y_1) \to A_2(x_2,y_2)\) and so on. Such that for \(A_k(x_k,y_k) \to A_{k+1}(x_{k+1},y_{k+1})\), exactly one of the following result holds:(i) \(x_{k+1} = x_k + a\) and \(y_{k+1} = y_k\)(ii) \(x_{k+1} = x_k\) and \(y_{k+1} = y_k + b\)for some \(a, b, k \in N\)If \(m = n = 6\), number of paths from \(A_0\) to \((m,n)\) consisting of exactly two perpendicular path with \(a, b \in \{1,2\}\).
6 balls marked as 1, 2, 3, 4, 5 and 6 are kept in a box. Two players A and B start to take out 1 ball at a time from the box one after another without replacing the ball till the game is over. The number marked on the ball is added each time to the previous sum to get the sum of numbers marked on the balls taken out. If this sum is even, then 1 point is given to the player. The first player to get 2 points is declared winner. At the start of the game, the sum is 0. If A starts to take out the ball, find the number of ways in which the game can be won.
If \(A = \{1, 2, 3, 4\}\), \(B = \{1, 2, 3, 4, 5, 6\}\) and \(f: A \to B\) is an injective mapping satisfying \(f(i) \neq i\) for all \(i \in A\), then number of such mappings are: