A cat is going up a stair well with sixteen stairs. However, instead of walking up the stairs one at a time, the cat jumps, going either two or three stairs up at each step (though if necessary, it will just walk that last step). Find the number different ways in which the cat can go from bottom to the top.

Question

A cat is going up a stair well with sixteen stairs. However, instead of walking up the stairs one at a time, the cat jumps, going either two or three stairs up at each step (though if necessary, it will just walk that last step). Find the number different ways in which the cat can go from bottom to the top. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Case I: When cat takes the last step as 1 Let

x

be the number of times she jumps 2 units Let

y

number of times she jumps 3 units

∴ 2 x + 3 y = 16 - 1 = 15

If y = 1, x = 6 \to 2222223 = 7

y = 3, x = 3 \to 222333 = \frac{6 !}{3 ! 3 !}

y = 5, x = 0 \to 33333 \to 1

Total in this cases

= 28

Case II: When she does not take the last step as unity

2 x + 3 y = 16

Hence, total

= 37 + 28 = 65