A method of breeding most commonly used when the desired variation is required to be induced in a vegetatively propagated crop is:

✅Explanation:
Vegetatively propagated crops, like potatoes, sugarcane, and bananas, are reproduced through asexual means like cuttings, grafting, or tissue culture. This means they retain the exact genetic makeup of the parent plant, limiting the natural variation available for selection in conventional breeding methods.

• Here's why the other options aren't the best choices:
● Polyploidy: While doubling the chromosome number can induce variation, it can also lead to instability and sterility in vegetatively propagated crops, making it less suitable.
● Pedigree method of breeding: This method relies on selecting and crossing individuals with desired traits over multiple generations. It's effective in self-pollinated crops but not ideal for vegetatively propagated ones due to their limited genetic variability.
● Backcross method of breeding: This involves repeatedly crossing a hybrid back with one of its parents to incorporate a specific trait. While useful for specific purposes, it's not the preferred method for inducing broad variation.
Mutation breeding: This method exposes seeds, buds, or tissues to radiation or chemicals to induce random mutations in the plant's DNA. While some mutations might be detrimental, others can lead to new and desirable traits. This unpredictable nature makes it a good option for introducing variation in vegetatively propagated crops where traditional breeding methods are limited.