Let \(S = \{1, 2, 3, \ldots, 9\}\). For \(k = 1, 2, \ldots, 5\), let \(N_k\) be the number of subsets of S, each containing five elements out of which exactly k are odd. Then \(N_1 + N_2 + N_3 + N_4 + N_5 =\)
Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is
An ordinary cubical dice having six faces marked with alphabets A, B, C, D, E, and F is thrown n times and the list of n alphabets showing up are noted. Find the total number of ways in which among the alphabets A, B, C, D, E, and F only three of them appear in the list.
A is a set containing n elements. A subset \(P_1\) of A is chosen. The set A is reconstructed by replacing the elements of \(P_1\). Next, a subset \(P_2\) of A is chosen and again the set is reconstructed by replacing the elements of \(P_2\). In this way, m (>1) subsets \(P_1, P_2, \ldots, P_m\) of A are chosen. The number of ways of choosing \(P_1, P_2, \ldots, P_m\) is
There are 5 points $P_1,P_2,P_3,P_4,P_5$ on the side $AB$, excluding $A$ and $B$, of a triangle $ABC$. Similarly there are 6 points $P_6,P_7,\ldots,P_{11}$ on the side $BC$ and 7 points $P_{12},P_{13},\ldots,P_{18}$ on the side $CA$ of the triangle. The number of triangles, that can be formed using the points $P_1,P_2,\ldots,P_{18}$ as vertices, is:
The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is