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The Diamond Resorts International® Mastercard® simply doesn't allow you to earn rewards for down payments, so that's a no. However, that's not the only option. You can choose any of the other credit cards that give you rewards in the manner of points, miles or cash back. Before paying down payments with a credit card, check that this option is available, otherwise use special services.
For example, the Discover it® Cash Back allows you to earn cash back on all purchases. Besides, you can match all earned cash back at the end of the year (if you are the new cardmember).