For n 2, let S_n denote the set of all subsets of 1,2, ,n with no two consecutive numbers.

Question

For $n\ge 2$, let ${S}_{n}$ denote the set of all subsets of ${1,2,\ldots ,n}$ with no two consecutive numbers. For example ${1,3,5}\in {S}_{6}$, but ${1,2,4}
otin {S}_{6}$. Then $n({S}_{5})$ is equal to \_\_\_\_\_\_\_\_

Answer: 13

Accepted Answer

To find $n({S}_{5})$, which is the number of subsets of ${1,2,3,4,5}$ with no consecutive numbers, we start by enumerating these subsets.

Let's denote the set ${1,2,3,4,5}$ as $A$. The subsets of $A$ that meet the criteria are:

The empty set: ${}$

Single-element sets: ${1}$, ${2}$, ${3}$, ${4}$, ${5}$

Two-element sets with no consecutive numbers: ${1,3}$, ${1,4}$, ${1,5}$, ${2,4}$, ${2,5}$, ${3,5}$

Three-element set with no consecutive numbers: ${1,3,5}$

Counting these subsets, we have:

1 subset with zero elements

5 subsets with one element

6 subsets with two elements

1 subset with three elements

Adding these counts, there are $1+5+6+1=13$ subsets in total.

Thus, $n({S}_{5})=13$.

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