in how many different orders can 4 children be arranged in a line,if john and julie refuse to stand next to one another ?
To find the number of different orders in which 4 children can be arranged in a line, we can start by considering all possible arrangements without any constraints. There are 4 children, so there are initially 4 choices for the first position, 3 choices for the second position, 2 choices for the third position, and 1 choice for the fourth position. Thus, there are 4 × 3 × 2 × 1 = 24 different orders in which the children can be arranged in a line without any restrictions.
However, we need to exclude the arrangements where John and Julie are standing next to each other.
To calculate this, we can treat John and Julie as a single entity. This means we are essentially arranging 3 entities (the pair of John and Julie, and the other two children) in a line. There are 3 choices for the first position (either John, Julie, or the other child), 2 choices for the second position, and 1 choice for the third position. Thus, there are 3 × 2 × 1 = 6 different orders in which the entities can be arranged.
Since John and Julie can swap places within the pair, we need to double the number of arrangements where they are standing next to each other. So, there are 2 × 6 = 12 arrangements with John and Julie standing next to each other.
Finally, we subtract this number from the total number of arrangements to get the number of different orders in which the 4 children can be arranged in a line with John and Julie not standing next to each other:
Total number of arrangements - Number of arrangements with John and Julie standing next to each other = 24 - 12 = 12 different orders.
Therefore, there are 12 different orders in which the 4 children can be arranged in a line without John and Julie standing next to each other.