101 Dalmations!

Cruella De Vil has 15 dalmations captive. She wishes to cage them into 5 distinct cages, with each cage containing 3 puppies. If each dalmatian is distinct, and the number of ways Cruella can arrange the 15 dalmatians in each of the cages is \(n\), find the sum of the digits of \(n\).

Image credit: Flickr PopCultureGeek
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