Find the number of sets A such that A ⊂ {1, 2, 4, 5, 6, 8, 9, 10, 11}, |A| = 4 and A contains no consecutive integers.

We are given a set $S=\{1,2,4,5,6,8,9,10,11\}$ and are tasked with finding how many subsets $A\subseteq S$ satisfy the following conditions:

• The size of $A$, $|A|=4$,
• $A$ contains no consecutive integers.

<h3 data-start="246" data-end="287">Step 1: Label the elements of the set</h3>

The elements of $S$ are $\{1,2,4,5,6,8,9,10,11\}$. Notice that there are 9 elements in total.

<h3 data-start="401" data-end="447">Step 2: Eliminate the consecutive elements</h3>

We need to select 4 elements from $S$ such that no two of them are consecutive. To simplify, let's examine the relative positions of these elements in the set.

• From the elements of $S$, we can see that $1,2$, $4,5,6$, and $8,9,10,11$ are groups of consecutive integers.
• The key is to treat each group of consecutive elements as blocks and select one element from each block, ensuring that we don't select two consecutive elements.

<h3 data-start="911" data-end="944">Step 3: Transform the problem</h3>

Let’s define a new set by removing the consecutive elements and instead consider their positions:

• $\{1,2\}$, $\{4,5,6\}$, and $\{8,9,10,11\}$.

These elements reduce the task to pick elements while ensuring that no two from different groups form a set with all subsets pleaseq then
By Sprunki

Posted by BevisJason on 2025-02-14 21:34:29