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**Hint:**Here we need to find compound interest that to compound annually, so we have to find out first years interest then second years compound interest then remaining years interest. By these steps we can find out the final answer.

**Complete step-by-step answer:**Given that $$P = 100000,\;r = 0.2 $$

(Converting $$2$$ years $$3$$ months to years) $$ = n = 2.25$$

Interest is compounded annually,

For first year $$ = 100000 \times 0.2 \times 1 = 20000$$

For second year $${P_{new}} = 120000$$

Interest $$ = 120000 \times 0.2 \times 1 = 24000$$

For last $$0.25\;$$ year, $${P_{new}} = 144000$$

Interest $$ = 144000 \times 0.2 \times 0.25 = 7200$$

New amount $$ = 144000 + 7200 = 151200$$

Interest $$ = 151200 - 100000 = 51200$$

Hence, the compound interest on Rs. $$100000$$ at $$20\% $$ per annum for $$2$$ years $$3$$ months, compound annually is $$ = 151200 - 100000 = 51200$$.

**Option C in correct answer**

**Note:**Usually on this type of question, we apply the formula for compound interest, but in this problem they asked to apply compound interest for $$2$$ years $$3$$ months, that too compounded annually, so we find compound interest for each year, then added them as principal amount for next year’s principle.

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