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I have a question about the dining plan (quick service) in Disney world located in Orlando.

I want to stay in a Hotel in Disney for 20 days, but I only want buy the dining plan for 15 days.

So can I buy the dining plan for 15 days and use it across 20 days? I’d like to do this because I have some lunches outside Disney; so the dining plan for 20 days will have meals that I won't use.

I couldn't find the information on the official website, however I found a blog post specifying all the rules on the dining plans. Specifically it says on dining plans:

Can I get the Dining Plan for more or fewer days than my hotel stay?

No. The Dining Plan can only be arranged for the exact number of
NIGHTS of your hotel package. The only way around this would be to
split your reservation. For example, if you’ll be staying at a WDW
hotel for five nights, but only want the Dining Plan for three nights
(because, say, you’re going to visit Universal Studios and Sea World),
you could make two different room/package reservations. One for three
nights and one for two. However, be aware that your Dining Plan
credits would only be valid for the package part of your stay. This
means that you would only be able to use the credits on the first
three consecutive days of your trip.

So it seems that you can use the credits freely within the period you have the dining plan for (you could use 3 meals on one day and only 1 the next day), but you can't use the credits for longer than the duration of the plan.

So can I buy the dining plan for 15 days and use it across 20 days?

No. The Disney Dining Plan is a lengh-of-stay option so unless you split your reservation, on or off property, you will get 20 night of credits.

But, it's still pretty easy to use them. If you have breakfast at your resort every day, that's half the credits right there. Or use a Snack Credit for a to-go item you can take to the other parks.

Then on non-Disney days, use one for lunch and dinner as well.

Also, keep in mind that you can use convert 1 Quick Service Credit to three Snack Credits which are often usable at the Epcot festival booths. It's very easy to burn through credits at Food & Wine or Flower & Garden.