Question:

In how many ways can $5$ children be arranged in a line such that (i) two particular children are always together (ii) two particular children are never together?

Updated On: Jul 6, 2022
  • $ 47, 73$
  • $48, 74$
  • $48, 72$
  • $49, 72$
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The Correct Option is C

Solution and Explanation

(i) We consider the arrangements by taking $2$ particular children together as one and hence the $4$ children can be arranged in $4! = 24$ ways. Again two particular children taken together can be arranged in two ways. Therefore, there are $24 \times 2 = 48$ total ways of arrangement. (ii) Among the $5! = 120$ permutations of $5$ children, there are $48$ in which two children are together. In the remaining $120 - 48 = 72$ permutations, two particular children are never together.
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Concepts Used:

Permutations and Combinations

Permutation:

Permutation is the method or the act of arranging members of a set into an order or a sequence. 

  • In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point. 
  • A permutation is used in many events of daily life. It is used for a list of data where the data order matters.

Combination:

Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.

  • Combination refers to the combination of about n things taken k at a time without any repetition.
  • The combination is used for a group of data where the order of data does not matter.