MathDB

We call a path Valid if i. It only comprises of the following kind of steps: A.

(x, y) \rightarrow (x + 1, y + 1)

(x, y) \rightarrow (x + 1, y - 1)

ii. It never goes below the x-axis. Let

M(n)

= set of all valid paths from

(0,0)

, to

(2n,0)

, where

n

is a natural number. Consider a Valid path

T \in M(n)

. Denote

\phi(T) = \prod_{i=1}^{2n} \mu_i

, where

\mu_i

= a)

1

, if the

i^{th}

step is

(x, y) \rightarrow (x + 1, y + 1)

y

, if the

i^{th}

step is

(x, y) \rightarrow (x + 1, y - 1)

Now Let

f(n) =\sum _{T \in M(n)} \phi(T)

. Evaluate the number of zeroes at the end in the decimal expansion of

f(2021)

3