Find the number of ways in which you can climb 13 steps if you can
go up 1 or 2 steps at a time.
You can take 1 single step and 6 double steps.
That's 7 steps and the positions for the single step
can be chosen in C(7,1) ways. That's 7 ways.
You can take 3 single steps and 5 double steps.
That's 8 steps and the positions for the single steps
can be chosen in C(8,3) ways. That's 56 ways.
You can take 5 single steps and 4 double steps.
That's 9 steps and the positions for the single steps
can be chosen in C(9,5) ways. That's 126 ways.
You can take 7 single steps and 3 double steps.
That's 10 steps and the positions for the single steps
can be chosen in C(10,7) ways. That's 120 ways.
You can take 9 single steps and 2 double steps.
That's 11 steps and the positions for the single steps
can be chosen in C(11,9) ways. That's 55 ways.
You can take 11 single steps and 1 double steps.
That's 12 steps and the positions for the single steps
can be chosen in C(12,11) ways. That's 12 ways.
You can take 13 single steps and 0 double steps.
That's 13 steps and the positions for the single steps
can be chosen in C(13,13) ways. That's 1 way.
Total: 7+56+126+120+55+12+1 = 377 ways.
Edwin
