I couldn't find my question. So...
How many subsets of the set {1,2,3,4,...10} have no two consecutive numbers as members?
+9665 If {a,b} and {b,a} is counted the same.......
Starting with 1:
{1,3}, {1,4}, ...... {1,9}, {1,3,5}, {1,3,6}...... ,{1,3,9}......
Subtotal 7+5+3+1 subsets = 16 subsets.
Starting with 2:
{2,4}, .......{2,9}, {2,4,6}, {2,4,7}... {2,4,9}...... Too lazy to type them all XD
Subtotal 6 + 4 + 2 = 12 subsets
Starting with 3:
{3,5}, {3,6} .... {3,9}, {3,5,7}, ..., {3 , 5, 9}.........
Subtotal 5 + 3 + 1 = 9 subsets
There we have found a rule for this.
Total subsets = 4 x 4 + 4 x 3 + 3 x 3 + 3 x 2 + 2 x 2 + 2 x 1 + 1 x 1 = 16 + 12 + 9 + 6 + 4 + 2 + 1 = 50 subsets.
