Brachylog, 18 bytes

{hl⟦kp;?z₀∋₍ᵐ+}ᶠot

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###Explanation

                ot      The output is the biggest result of…
{             }ᶠ        …finding all outputs to the following predicate:
 hl⟦k                     Construct the range [0, …, n-1]
     p                    Take a permutation of that range
      ;?z₀                Zip that permutation with the Input, stopping when all elements of
                            the input are used (important because the range is possibly
                            bigger than the length of the input)
          ∋₍ᵐ             In each element of the zip, take the head'th element of the tail
             +            Sum the result

Brachylog, 18 bytes

{hl⟦kp;?z₀∋₍ᵐ+}ᶠot

Try it online!

Explanation

                ot      The output is the biggest result of…
{             }ᶠ        …finding all outputs to the following predicate:
 hl⟦k                     Construct the range [0, …, n-1]
     p                    Take a permutation of that range
      ;?z₀                Zip that permutation with the Input, stopping when all elements of
                            the input are used (important because the range is possibly
                            bigger than the length of the input)
          ∋₍ᵐ             In each element of the zip, take the head'th element of the tail
             +            Sum the result