1. Interpret the problem

A child buys $6$ ice-creams from $5$ available types.

Let the numbers bought of the five types be $x_1,x_2,x_3,x_4,x_5$. Then $x_1+x_2+x_3+x_4+x_5=6$ where each $x_i\ge 0$ is an integer.

So we need the number of non-negative integer solutions of this equation.

2. Count the number of ways using stars and bars

The number of non-negative integer solutions of $x_1+x_2+x_3+x_4+x_5=6$ is $\binom{6+5-1}{5-1}=\binom{10}{4}=\binom{10}{6}=210.$

So the correct number of ways is not $\binom{10}{5}$.

Since $\binom{10}{5}=252 \neq 210,$ Statement 1 is false.

3. Check Statement 2

Statement 2 says this number equals the number of arrangements of $6A$ and $4B$ in a row.

The number of arrangements of $6A$ and $4B$ is $\frac{10!}{6!4!}=\binom{10}{4}=210.$

This matches the required count.

Why does this correspondence work?

In stars and bars, represent the $6$ ice-creams by $6$ A's and the $4$ separators between $5$ types by $4$ B's. For example, $AA\,B\,A\,BB\,AAA\,B$ would mean counts distributed among the five types.

Thus the number of ways to buy the six ice-creams is exactly the number of arrangements of $6A$ and $4B$.

Hence Statement 2 is true.

4. Evaluate the options

• Statement 1: False
• Statement 2: True

Therefore, the correct option is: $\boxed{A}$

5. Compare with stored answer

Stored correct answer: $A$

My derived answer: $A$

They agree.