MongoDB Sorting by relevance by matching sub documents

I have a document that must be searched according to a value in its subdocument and sorted by relevancy.

While researching this topic, I came across this question. I now know I have to use the aggregation framework, but the solution didn't cover subdocuments.

For example purposes say the I wanted to grab every document that had sub documents with the teams.name ['blue', 'red', 'green'] and sorted based on the most matches it had.

League Collection:
{ 
  coach: Henderson,
  teams:
  [ 
    {
       name: red,
       logo: bird,
       division: east
    },
    {
       name: blue,
       logo: bluejay,
       division: west
    },
   {
       name: green,
       logo: monkey,
       division: east
    }
 ]
}

{ 
  coach: Wilkins,
  teams:
  [ 
    {
       name: red,
       logo: bird,
       division: east
    },
    {
       name: blue,
       logo: bluejay,
       division: west
    },
 ]
}

{ 
  coach: Sandy,
  teams:
  [ 
    {
       name: red,
       logo: bird,
       division: east
    }
 ]
}

The end result would be:

item          total matches
--------      --------------
Henderson     3
Wilkins       2
Sandy         1

How would I achieve this?