• # question_answer In a small village, there are 87 families, of which 52 families have at most 2 children. In a rural development programme 20 families are to be chosen for assistance, of which at least 18 families must have at most 2 children. In how many ways can the choice be made? A) $^{52}{{C}_{18}}^{35}{{C}_{2}}$ B) $^{52}{{C}_{18}}{{\times }^{35}}{{C}_{2}}{{+}^{52}}{{C}_{19}}{{\times }^{35}}{{C}_{1}}{{+}^{52}}{{C}_{20}}$ C) $^{52}{{C}_{18}}{{+}^{35}}{{C}_{2}}{{+}^{52}}{{C}_{19}}$ D) $^{52}{{C}_{18}}{{\times }^{35}}{{C}_{2}}{{+}^{35}}{{C}_{1}}{{\times }^{52}}{{C}_{19}}$

[b] The following are the number of possible choices: $^{52}{{C}_{18}}{{\times }^{35}}{{C}_{2}}$ (18 families having at most 2 children and 2 selected from other type of families) $^{52}{{C}_{19}}{{\times }^{35}}{{C}_{1}}$(19 families having at most 2 children and 1 selected from other type of families) $^{52}{{C}_{20}}$ (All selected 20 families having at most 2 children). Hence, the total number of possible choices is: ${{=}^{52}}{{C}_{18}}{{\times }^{35}}{{C}_{2}}{{+}^{52}}{{C}_{19}}{{\times }^{35}}{{C}_{1}}{{+}^{52}}{{C}_{20}}$