An ice-cream shop sells $11$ kinds of ice-cream, including mango and lemon. An ice-cream  shop sells $11$ kinds of ice-cream, including mango and lemon. For a bowl, one chooses at random five kinds (not necessarily different).

$(a)$ How many different bowls can be made?  $A=11 \cdot 11\cdot 11\cdot 11\cdot 11$

$(b)$ How many different bowls contain both lemon and mango? $A=1 \cdot 1\cdot 11\cdot 11\cdot 11$

$(c)$ How many bowls contain lemon, but not mango?
 $A=1 \cdot 10\cdot 10\cdot 10\cdot 10$

I don't know if my answers are correct. Can someone explain me this problem please? Thank you.
You are a bit off.

Let's go with order.

The main question is whether the order of the selection is important or not. That is to say, is a [Mango Mango Mango Lemon Lemon] bowl different than a [Lemon Lemon Mango Mango Mango] one? Let's see how this changes the things

Question 1: How many possible bowls can exist?

If the order is important, you are right. There are 11^5 possibilities. More specifically, this is called Permutations (with repetitions) of 11 elements in class 5. In general, the formula for P(n, k) = n^k.

If the order does not matter, then we need to consider how many of those 11^5 are actually duplicated. This is called Combinations (with repetitions) of 11 elements in class 5. In general, the formula for C(n, k) is (n+k-1)! / (k! * (n-1)!) , so in your case that is 15! / (5! * 10!)

Question 2: How many contain both lemon and mango?
Well, we need at least 1 mango and 1 lemon, so we can lock those down and then consider the rest as either:
P(11, 3) or C(11, 3) (2 flavors are locked, the rest can be any)

Question 3: How many contain lemon but not mango?

Again, we can lock down the lemon, but the other flavors must be anything but mango, so it becomes
P(10, 4) or C(10, 4) (we remove mango from the options for the remaining 4 flavors)
You are right if you consider mango and lemon to be different from lemon and mango. I mean to say, If the order of adding the different components to the ice-cream matters