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I got a sigma notation expression:
Σ nCr(10-n, n)
In this case, n denotes the number of leaps of 2 steps, which leaves the number of single step to be (10-2n), and the total number of steps to be (10-2n)+n=10-n, among which there are n two-step leaps, and the number of different arrangements of such steps being nCr(10-n, n). Adding together the different number of ways this can be done (ranging from no two-step leaps to a max of 5 two-step leaps for a 10-step staircase) should give the answer.