1. Recall the definition of a transition element
   A transition element is an element whose atom or common ion has a partially filled $d$-subshell. In practice, these are generally the elements of the $d$-block.

2. Check each atomic number option-wise

Option A: $21, 32, 53, 64$

• $21$ = Scandium $(\mathrm{Sc})$ → transition element
• $32$ = Germanium $(\mathrm{Ge})$ → $p$-block, not transition
• $53$ = Iodine $(\mathrm{I})$ → halogen, not transition
• $64$ = Gadolinium $(\mathrm{Gd})$ → inner transition element ($f$-block), not counted among ordinary transition elements here

So, Option A is not correct.

Option B: $9, 17, 34, 38$

• $9$ = Fluorine $(\mathrm{F})$ → non-metal, not transition
• $17$ = Chlorine $(\mathrm{Cl})$ → non-metal, not transition
• $34$ = Selenium $(\mathrm{Se})$ → $p$-block, not transition
• $38$ = Strontium $(\mathrm{Sr})$ → $s$-block, not transition

So, Option B is not correct.

Option C: $37, 42, 50, 64$

• $37$ = Rubidium $(\mathrm{Rb})$ → $s$-block, not transition
• $42$ = Molybdenum $(\mathrm{Mo})$ → transition element
• $50$ = Tin $(\mathrm{Sn})$ → $p$-block, not transition
• $64$ = Gadolinium $(\mathrm{Gd})$ → inner transition element

So, Option C is not correct.

Option D: $21, 25, 42, 72$

• $21$ = Scandium $(\mathrm{Sc})$ → transition element
• $25$ = Manganese $(\mathrm{Mn})$ → transition element
• $42$ = Molybdenum $(\mathrm{Mo})$ → transition element
• $72$ = Hafnium $(\mathrm{Hf})$ → transition element

All four are transition elements.

So, Option D is correct.

3. Final Answer $\boxed{\text{D: } 21, 25, 42, 72}$

4. Comparison with stored correct answer
   Stored correct answer = D.
   My derived answer = D.
   Hence, they agree.