For n 2, let Sn denote the set of all subsets of 1, 2, 3,, n with no two consecutive numbers. For example, 1, 3, 5 S6, b

Question

For n 2, let Sn denote the set of all subsets of 1, 2, 3,, n with no two consecutive numbers. For example, 1, 3, 5 S6, but 1, 2, 4 S6. Then, find n(S5).

Accepted Answer

Given: We need the number of subsets of $\{1, 2, 3, 4, 5\}$ such that no two selected elements are consecutive.

Find: $n(S_5)$ .

Define $a_n$ as the number of valid subsets of $\{1, 2, \ldots, n\}$ with no two consecutive elements.

If $n$ is not included, then the subset is chosen from $\{1, 2, \ldots, n-1\}$ , giving $a_{n-1}$ possibilities.

If $n$ is included, then $n-1$ cannot be included, so the remaining subset is chosen from $\{1, 2, \ldots, n-2\}$ , giving $a_{n-2}$ possibilities.

Therefore,

a_n = a_{n-1} + a_{n-2}

Using the base cases from the extracted solution:

a_2 = 3, \qquad a_3 = 4

Now,

a_4 = a_3 + a_2 = 4 + 3 = 7

and

a_5 = a_4 + a_3 = 7 + 4 = 13

Therefore, $n(S_5) = 13$ .

Given: The set is A = 1, 2, 3, 4, 5. Find: The number of subsets with no two consecutive elements. For subsets having exactly r elements with no two consecutive elements, the count is ^n-r+1Cr For n = 5, this becomes ^6-rCr Now count for all possible values of r: - No element: 1 way - Exactly 1 element: ^5C1 = 5 - Exactly 2 elements: ^4C2 = 6 - Exactly 3 elements: ^3C3 = 1 Hence, 1 + 5 + 6 + 1 = 13 Therefore, the total number of such subsets is 13.

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