To solve this problem, we need to analyze the nuclear reactions step-by-step.$\\$The initial nucleus is $^{200}_{82}X. \\$1. Alpha decay:$\\$In alpha decay, an alpha particle $(^{4}_{2}\alpha)$ is emitted.$\\$This reduces the mass number by 4 and the atomic number by 2.$\\ ^{200}_{82}X \xrightarrow{\alpha} ^{196}_{80}Y \\$2. Positron emission:$\\$In positron emission, a positron $(e^+)$ is emitted.$\\$This reduces the atomic number by 1.$\\ ^{196}_{80}Y \xrightarrow{e^+} ^{196}_{79}Z \\$3. Beta-minus decay:$\\$In beta-minus decay, a beta particle $(\beta^-)$ is emitted.$\\$This increases the atomic number by 1.$\\ ^{196}_{79}Z \xrightarrow{\beta^-} ^{196}_{80}P \\$4. Electron capture:$\\$In electron capture, an electron $(e^-)$ is captured by the nucleus.$\\$This reduces the atomic number by 1.$\\ ^{196}_{80}P \xrightarrow{e^-} ^{196}_{79}Q \\$Thus, the mass number and atomic number of the product $Q$ are:$\\$Mass number: $196 \\$Atomic number: $79 \\$The correct option is not listed among the given options, indicating a potential error in the options provided.$\\$However, based on the calculations, the mass number is 196 and the atomic number is 79.