To write mathematica code for summation using a given data

I am given a set like some terms from the partition of $20$:

${{17, 3}, {13, 7}, {11, 3, 3, 3}, {7, 7, 3, 3}, {7, 5, 5, 3}, {5, 5,
5, 5}}$.

From the above data,

$20= underline{1}*17+underline{1}*3= underline{1}*13+underline{1}*7 =underline{1}*11+underline{3}*3 = underline{2}*7+underline{2}*3 = underline{1}*7+underline{2}*5+underline{1}*3 = underline{4}*5$

Now the above underlined numbers indicate the number of repetitions of each term appearing in the partition. I need to write those number of repetitions in the form of summation involving factorials:

Like for the above case, I need to represent the summation for the above partition as:

${T=frac{1}{1!*1!}+frac{1}{1!*1!}+frac{1}{1!*3!}+frac{1}{2!*2!}+frac{1}{1!*2!}+frac{1}{4!}}$

How can I do that?

fn=Tr[1/Times @@@ (Tally[#][[All, 2]] & /@ #!)]&;

fn@{{17, 3}, {13, 7}, {11, 3, 3, 3}, {7, 7, 3, 3}, {7, 5, 5, 3}, {5, 5, 
  5, 5}}

ip = {{17, 3}, {13, 7}, {11, 3, 3, 3}, {7, 7, 3, 3}, {7, 5, 5, 3}, {5, 5, 5, 5}};

Total[Times @@ Values[1 / Factorial @ #] & /@ Counts /@ ip]

Query[Total, Times @@ Values[1/Factorial @ #] &] @ Dataset[Counts /@ ip]

Dataset[Counts /@ ip][Total, Times @@ Values[1/Factorial @ #] &]