Permutations And Combinations Question 340

Question: 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?

Options:

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} $

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Answer:

Correct Answer: B

Solution:

  • [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} $