Answer: A

The possible selections of three different coins from $1$, $2$, $5$, $10$ and $20$ are:

$1+2+5=8$

$1+2+10=13$

$1+2+20=23$

$1+5+10=16$

$1+5+20=26$

$1+10+20=31$

$2+5+10=17$

$2+5+20=27$

$2+10+20=32$

$5+10+20=35$

Statement I: $m$ is not a prime number.

The possible non-prime values of $m$ are $8$, $16$, $26$, $27$, $32$ and $35$.

Here, $8$, $16$, $26$, $27$ and $35$ involve a coin of denomination $5$, but $32$ comes from $2+10+20$, which does not involve a coin of denomination $5$.

So, Statement I alone is not sufficient.

Statement II: The sum of the digits of $m$ is greater than $5$.

The values of $m$ whose digit sum is greater than $5$ are $8$, $16$, $17$, $26$, $27$ and $35$.

Each of these sums is obtained from a combination that includes the coin of denomination $5$.

So, Statement II alone is sufficient to answer the question.

Therefore, the question can be answered using one statement alone, but not using the other statement.