## Task 3 - 211213

A set $M$ contains exactly 55 elements. For each natural number $k$ with $0 \leq k \leq 55$, let $A_{k}$ denote the number of all those subsets of $M$ that contain exactly $k$ elements.

Determine all those natural numbers $k$ for which $A_{k}$ is the largest!
I. If 3 is a natural number with $0 \leq k \leq 55$, then the number $A_{k}$ is determined as follows:

Imagine each of the mentioned subsets is formed by sequentially selecting a first, a second, ..., and a $k$-th element from $M$. For the first element, there are exactly 55 possible choices. After each such selection, there are exactly (55-1) possibilities for the second element, and so on.

Finally, after each possibility of selecting $k-1$ elements, there are exactly $(55-(k-1))$ possibilities for the $k$-th element. Thus, there are a total of $55 \cdot 54 \cdot \ldots \cdot(56-k)$ possibilities for sequentially selecting $k$ elements in the described manner.

Of these selection possibilities, exactly those that differ only in the order of the selected $k$ elements lead to the same subset. For the selection of such an order, there are exactly $k \cdot(k-1) \cdot \ldots \cdot 1$ possibilities (since in each such order, one of the $k$ elements can be placed in the first position, one of the remaining $(k-1)$ elements in the second position, and so on, ..., until for the $k$-th position, exactly 1 element remains in each already fixed order of the first $(k-1)$ elements). Therefore,

$$
A_{k}=\frac{55 \cdot 54 \cdot \ldots \cdot(56-k)}{1 \cdot 2 \cdot \ldots \cdot k}
$$

II. If $k$ is a natural number with $0 \leq k \leq 55$, then correspondingly

$$
A_{k+1}=\frac{55 \cdot 54 \cdot \ldots \cdot(56-k) \cdot(55-k)}{1 \cdot 2 \cdot \ldots \cdot k \cdot(k+1)} \quad \text { so it follows that } \quad A_{k+1}=\frac{55-k}{k+1} \cdot A_{k}
$$

III. For $k \leq 26$, the fraction $\frac{55-k}{k+1}$ is greater than 1, for $k=27$ it is equal to 1, and for $28 \leq k \leq 55$ it is less than 1 but positive. Therefore,

$$
A_{0}A 29>\ldots>A_{54}>A_{55}
$$

Thus, $A_{k}$ is maximized exactly for $k=27$ and $k=28$.

Adapted from $[5]$