solve quick ## Number of Mappings, Injections, Surjections### a. Counting Total Number of Mappings from A to B (|A| = n, |B| = m) in Two Ways:\[ m^n = \left( \binom{m}{1} \right) \ast s(n, 1) \ast 1! + \left( \binom{m}{2} \right) \ast s(n, 2) \ast 2! + \cdots + \left( \binom{m}{m} \right) \ast s(n, m) \ast m!\]- **Explanation**: This equation represents the total number of mappings from set A to set B, sliced by the increasing size of the image of the function. You must achieve the same answer using this method.- **Verification**: Verify this identity for n = 7, m = 4 using Pascal and Stirling triangles.### b. Number of Surjections (Onto Maps)- **Task**: Give the number of surjections from A to B (|A| = 3, |B| = 5).### c. Number of Injections (One-to-One Maps)- **Task**: Find the number of injections from A to B (|A| = 3, |B| = 5).### Additional Notes- **Pascal's Triangle**: A triangular array used to calculate combinations.- **Stirling Numbers**: A set of numbers used in combinatorics to count partitions of a set.This content explores advanced combinatorial concepts, useful in mathematical problem-solving and analysis.

sterling's table: k→ n|⏟ 0 1 2 3 4 5 6 7 8 9 10 0 1 1 0 1 2 0 1 1 3 0 1 3 1 4 0 1 7 6 1 5 0 1 15 25 10 1 6 0 1 31 90 65 15 1 7 0 1 63 301 350 140 21 1 8 0 1 127 966 1701 1050 266 28 1 9 0 1 255 3025 7770 6951 2646 462 36 1 10 0 1 511 9330 34105 42525 22827 5880 750 45 1 Pascals triangle: