Question

how many different 2 or 4 number sequences can be chosen from the set (1,2,3,4,5,6,7,8,9,10)?

9. How many different 2 or 4 number sequences can be chosen from the set (1,2,3,4,5,6,7,8,9,10)?

Answer 1

Common Difference	Sequence of two numbers	Total
1	{(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9),(9,10)}	9
2	{ (1,3), (2,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10)}	8
3	{(1,4),(2,5),(3,6),(4,7),(5,8),(6,9),(7,10)}	7
4	{(1,5),(2,6),(3,7),(4,8),(5,9),(6,10)}	6
5	{(1,6),(2,7),(3,8),(4,9),(5,10)}	5
6	{(1,7),(2,8),(3,9),(4,10)}	4
7	{(1,8),(2,9),(3,10)}	3
8	{(1,9),(2,10)}	2
9	{(1,10)}	1
Total		45

Common Difference	Sequence of 4 number	Total
1	{(1,2,3,4),(2,3,4,5),(3,4,5,6),(4,5,6,7),(5,6,7,8),(6,7,8,9),(7,8,9,10)}	7
2	{(1,3,5,7),(2,4,6,8),(3,5,7,9),(4,6,8,10)}	4
3	{(1,4,7,10)}	1
Total		12

Hence Total number of ways to create sequence of different 2 and 4 numbers is

= 45 + 12 = 57 .