Replace the underlined words with an Idiom from the following options :

I suspect that the thief has sent the policemen on a foolish and fruitless search.

a wild - goose chase

The correct idiom to replace the underlined words is: a wild-goose chase

The phrase "a wild-goose chase" is used to describe a search that is futile or hopeless. In the sentence "I suspect that the thief has sent the policemen on a foolish and fruitless search," the speaker is suggesting that the thief has deliberately misled the police by sending them on a search that is unlikely to be successful.