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Mapping Problem

Source: 2002 National High School Mathematics League, Exam One, Problem 5

March 13, 2020

Problem Statement

Two sets of real numbers A={a1,a2,⋯ ,a100},B={b1,b2,⋯ ,b50}A=\{a_1,a_2,\cdots,a_{100}\},B=\{b_1,b_2,\cdots,b_{50}\}. Mapping f:A→Bf:A\to B, ∀i(1≤i≤50),∃j(1≤j≤100),f(aj)=bi\forall i(1\leq i\leq 50),\exists j(1\leq j\leq100),f(a_j)=b_i, and f(a1)≤f(a2)≤⋯≤f(a100)f(a_1)\leq f(a_2)\leq\cdots\leq f(a_{100}) Then the number of different ff is (A)C10050(B)C9950(C)C10049(D)C9949\text{(A)}\text{C}_{100}^{50}\qquad\text{(B)}\text{C}_{99}^{50}\qquad\text{(C)}\text{C}_{100}^{49}\qquad\text{(D)}\text{C}_{99}^{49}
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