List of top Mathematics Questions on permutations and combinations asked in JEE Advanced

Let \[ S=\{1,2,3,\ldots,10\}. \] Consider the set \[ X=\{R : R \text{ is an equivalence relation on } S \text{ such that } R \text{ has exactly } 42 \text{ elements}\}. \] Then the number of elements in \(X\) is __________.
Let \(\mathbb{N}\) denote the set of all positive integers. Consider the sets \[ A = \{1,2,3,4,5\} \] and \[ B = \{1,2,3,4,5,6,7\}. \] Let \(S\) be the set of all functions \[ f : A \to B \] such that \[ f(2) \ne 2 \quad \text{and} \quad f(4) \ne 4. \] Consider the set \[ T = \left\{ f \in S : \text{there exists a function } g : B \to \mathbb{N} \text{ such that } g(f(x)) = 2^x \text{ for all } x \in A \right\}. \] Then the number of elements in the set \(T\) is:
The number of ways to distribute \(10\) identical red pens and \(14\) identical blue pens among four persons such that each person gets \(6\) pens is ________.