Consider all functions \(f : \{1, 2, 3, 4\} \to \{1, 2, 3, 4\}\) which are one-one, onto and satisfy the following property: if \(f(k)\) is odd then \(f(k+1)\) is even, \(k = 1, 2, 3\). The number of such functions is:
Let $ABC$ be a triangle. Consider four points $p_1,p_2,p_3,p_4$ on side $AB$, five points $p_5,p_6,p_7,p_8,p_9$ on side $BC$, and four points $p_{10},p_{11},p_{12},p_{13}$ on side $AC$. None of these points is a vertex of the triangle. Then the total number of pentagons that can be formed by taking all the vertices from the points $p_1,p_2,\ldots,p_{13}$ is _____.
Let $S=\{(m,n):m,n\in\{1,2,3,\ldots,50\}\}$. If the number of elements $(m,n)$ in $S$ such that $6^m+9^n$ is a multiple of 5 is $p$ and the number of elements $(m,n)$ in $S$ such that $m+n$ is a square of a prime number is $q$, then $p+q$ is equal to _____.
The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits $0,1,2,5,9$, if the repetition of the digits is allowed, is _____.
Let x be the elements of the set A = \{1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120\} and x_1, x_2, x_3 be positive integers and d be the number of integral solutions of x_1 x_2 x_3 = x, then d is
Five different digits from the set of numbers \(\{1, 2, 3, 4, 5, 6, 7\}\) are written in random order. How many numbers can be formed using 5 different digits from this set if the number is divisible by 9?